I’ve written a number of posts about methane and have also described an updated global warming potential metric, typically called GWP*. I happen to quite like this updated metric, but I noticed a tweet suggesting that it was

*designed* to benefit high-emitting countries and industries.

I found this suggestion a little odd, and posted a tweet thread of my own. However, since I have a blog, I thought I could turn it into a blog post too 🙂

A few years ago I was somewhat cofused about why there was so much focus on methane. If it has a relatively short atmospheric lifetime (~12 years) then it shouldn’t accumulate in the same way as CO_{2} and should have a different impact on long-term warming.

I tried asking people about this, but couldn’t really resolve my confusion. I even engaged in an email exchange with colleagues at my university, but they didn’t seem to understand what I was asking and just repeated the standard GWP values.

I then came across work by Michelle Cain, which described a new metric called GWP*. This new metric seemed to do a much better job of representing the warming due to a short-lived greenhouse gas like methane than the more standard GWP100, or GWP20, metrics. If you want to understand this new metric, this Carbon Brief article is very good.

As someone who had been confused about the focus on methane, this new metric seemed to be an obvious update that largely resolved a flaw with the standard GWP100, or GWP20, metrics (which don’t really distingush between stock and flow pollutants). It seemed much more likely that this work was motivated by trying to solve this issue, than it being **designed* to benefit high-emitting countries and industries.*

However, it is the case that some could use this new metric to claim that they were no longer contributing to global warming because, for example, they were no longer increasing their methane emissions. Also, methane is responsible for quite a lot of warming (~0.5^{o}C if you include the direct and indirect effects). If methane emissions are not reduced, then this warming will persist and this will make it extremely difficult to meet some of the targets.

However, from my perspective, if we want to use metrics to estimate the impact of different greenhouse gases, then it seems preferable to use ones that do a reasonable job of this, than to stick with ones that have flaws, even if such an update may be misused by some to argue against reducing their emissions.

Also, I can’t really see any reason why GWP* couldn’t be used to demonstrate how different industries/countries have contributed to historical warming, how they might contribute to future warming, and how future emission cuts could be apportioned. The standard GWP metrics might be easier, but that doesn’t necessarily imply better.

**Links:**

Posts I’ve written about methane.

Posts I’ve written about GWP*

Guest post: A new way to assess ‘global warming potential’ of short-lived pollutants — Carbon Brief article about the new GWP* metric.

Improved calculation of warming-equivalent emissions for short-lived climate pollutants — Michelle Cain et al’s paper about GWP*.

I think it would be fairer to say that GWP* was designed by people who thought standard metrics overemphasise the importance of methane. Many of the chief proponents work in the agriculture sector and research organisations aligned with agriculture, unsurprisingly.

We sort of already had this debate with GWP20 vs. GWP100, and those who think methane is relatively unimportant already got their way, so I think there is some frustration that people want to relitigate this issue.

Mostly what a metric does is give you a single figure of merit that allows you to compare how bad two gases are for the climate, allowing you to ignore details about where and when the effect takes place.

From this point of view, GWP* is not a metric at all. It is an ‘anti-metric’, whose proponents oppose the idea of averaging out the time dependence to produce a simple figure of merit.

Of course, it is a neat way of producing a fairly accurate time-profile of forcing, although it would be a lot less complicated to just say ‘warming from methane is proportional to emissions in the last 20 years, but warming from CO2 is proportional to emissions since the dawn of history’.

And the extra complication allows e.g. the cattle industry to misleadingly claim they are causing ‘zero emissions’ in GWP* terms (they also tend to elide N2O emissions and land use issues). Sometimes simple and inaccurate is better than accurate but complicated and confusing.

Did the realclimate post on methane from Gavin help resolve your confusion regarding methane? I thought it was pretty well presented and persuasive.

https://www.realclimate.org/index.php/archives/2021/09/the-definitive-co2-ch4-comparison-post/

Cheers

Mike

Ben,

I guess many of the proponents work in the agricultural sector, but I’m not aware of any such association between the researchers in the UK and the agricultural sector. I agree that GWP* isn’t really a metric, since it ends up being more complicated than just a single number. I kind of agree about simple but inaccurate potentially being better than accurate and confusing, but I do share the concerns that some have in using a metric that makes methane seem more like a stock pollutant than a flow pollutant. This would be fine if everyone was aware of the distinction, but I’m not sure all are.

Ben sez:

“I think it would be fairer to say that GWP* was designed by people who thought standard metrics overemphasise the importance of methane.”

No it wasn’t.

Ben sez:

No it wasn’t.

“It was designed to capture warming from a time-series of greenhouse gases. Here are the key papers:”

(hopefully two back-to-back tweets appear👇)

“Let’s see the evidence that it was designed to benefit anyone?”

(the two quotes are via David Frame)

And although “designed by” and “proponents of” are different descriptions of people’s roles, you would be hard-pressed to show (because it’s false) that the people (authors above) who developed the concept were a priori deliberately trying to favour particular gases or forcings or sectors.

So, my point being – and the point of ATTP’s original post – is that GWP* gives a better description of ultimate temperature impacts of various forcing species – especially if you combine with historical and/or future emissions pathways.

From a strictly physical perspective – forgetting about cows and the people who love cows, etc. – GWP* is simply more coherent.

And since with methane, we are trying to figure out both the ultimate and peak *temperature* impacts, it’s best to be using a metric that actually does this.👇

What will Australia’s Scott Morrison make of it? Curbing methane may be a hard sell in a nation whose PM once said ” Coal is now an essential component of our zero carbon future.”

I think a ‘social cost of CH4’ versus ‘social cost of CO2’ would work better and at least not involve weird time dependence issues, negative values, and grandfathering problems like GWP*. Ultimately, metrics are economics/policy concepts, not physics ones. And people who aren’t physicists need to understand them.

Or just use GWP100, and get on with the job, would also be perfectly sufficient for the next couple of decades.

Also, as ATTP+rust pointed out, the people who came up with GWP* were principally physicists. What I wrote was misleading because it didn’t distinguish two overlapping sets clearly enough: the people who came up with the idea, and those who are the current-day advocates for it.

I think the question of whether GWP* was designed in good faith is an unhelpful distraction, what matters is how it gets (mis)used.

Ben,

Indeed, I was mostly responding in the post to the suggestion that it was *designed* to help high-emitters, which seems an odd claim. I completely agree that it could (and already is) be used to argue against making cuts to methane emissions. It may be that sticking with GWP could have avoided that. On the other hand, if people became aware of the potential with using GWP, that could backfire. You also have the issue that GWP could potentially imply an equivalence between cuts to methane emissions and cuts to CO2 emissions.

I think that e.g. the kind of thing Paul Price was doing was explicitly looking at how to reduce both CO2+CH4 emissions in Ireland to meet carbon targets using GWP*, which of course is directly trading one gas off against the other.

So changing the metric doesn’t prevent ‘trade-offs’; of course, you can argue that you are then at least making the correct trade-off.

I suspect that using GWP* and some discounting (~1%) would actually give you a similar answer in terms of ‘optimal trade-off’ between CO2 and CH4 in economic terms as GWP100. You can still do silly things like kick the can down the road in the hope that NETs become available.

In practice, though, arguing against bringing in CH4 pricing using GWP100 does prevent any trade-offs happening in the short term, though, because it frustrates the creation of a market in CH4 reduction. i.e. preserves the status-quo of no action on CH4 at all.

Ben,

Indeed, and my argument is certainly not as simple as “don’t use GWP100”. My point is mostly that it would be good if people were aware of this issue, even if they do still use GWP100.

I think the best would be to think of forcers as two different kinds, long lived and short lived. The former once emitted are there to stay, and we cannot do anything about it later. The short lived can be dealt with sooner if it doesn’t cost much, and doesn’t distract from reducing emissions of the long lived, or later if it does, and it is by then deemed worthwhile.

The issue with methane is that many of the emissions reductions actually distract in one way or the other from efforts to reduce fossil fuel emissions, be it through trading schemes or plain simple public frustration. But reducing methane emissions at the expense of carbon dioxide is not going to stabilise climate. On the contrary you will end up with more long term warming.

I realise I am going out on a limb here, but this is why I don’t think any of these metrics are particularly useful if what you are after is to stabilise climate.

Thorsten,

Yes, I agree that it would probably be better to simply regard long-lived forcers and short-lived forcers separately and that it’s important to realise that once a long-lived forcer has been emitted there is nothing we can do later (absent some kind of negative emission technology) while a delay to the reduction in emissions of a short-lived forcer doesn’t carry this risk.

A point I’ve made myself on a number of occasions 🙂

Yes, this may be a fair point. I do think GWP* does a reasonable job of linking emissions to warming, but you may be right that none of these metrics are particularly useful if the goal is to stabilise global warming.

Thorsten: aren’t you misrepresenting things when you say the long lived forcers …

“once emitted are there to stay, and we cannot do anything about it later.”

Capture of long lived forcers is clearly referenced in the IPCC planning, is it not? It’s a bit of a fine point, but I think maybe what you mean to say is that we won’t like the cost of addressing the long lived forcers after they have been emitted, right?

Cheers

Mike

Mike,

Yes, I suspect that Thorsten is aware of negative emissions technologies. They may well play a role, but it’s not yet clear if they can really work at a suitable scale. Also, bear in mind that there is a difference between CCS (carbon capture and storage) and carbon dioxide removal (CDR). In the former, carbon dioxide is captured at source and stored. This, however, cannot be a negative emission (i.e., you can’t capture more than is emitted). The only way to get to negative emissions is to find some way to capture carbon dioxide from the atmosphere and store it.

I don’t keep track of all the details, but I believe what you say about the NE technologies. I think that our success at managing the warming requires that these technologies work at scale. Maybe I am mistaken about that?

If I am right about that then I think we need to accurately about our various current emissions, how we reduce them, or how we kick them off into the future task of bring the NE technologies online and up to scale.

I don’t think there are any realistic plans for managing global warming that don’t include the NE technologies. Am I right or wrong about that? If right, then we should have the NE technology explicitly in discussions about our future. Does that seem reasonable?

“The only way to get to negative emissions is to find some way to capture carbon dioxide from the atmosphere and store it.”

Some of it will inevitably re-captured. CO2 gas wells are a thing, and a major high-pressure CO2 gas field in New Mexico was tapped and connected to California during WWII, when the war in the South Pacific created an instant need for dockside and shipboard refrigeration.

There is a stunning lack of postwar reports on how , or if , the wells were capped and managed .

smallbluemilk, yes I am aware of negative emissions technologies, and from what I have seen thus far, it is all unicorns. The cost of removing directly from the atmosphere is immense. Think of it, in order to lower from say 600 to 400 ppm, you need a fan that will blow 1/3 of the atmosphere into your removal gizmo, and that is if it was to be 100 percent effective. Otherwise, you need a bigger fan. The number I saw from the Icelandic facility was about 1000 $/tonne, well above the EU emissions trading level. Alternatively you use large forests, chop down the trees and store them, or their carbon at least, in order to accelerate the natural removal process. But given how people are already fighting over forests, for many good reasons, I don’t see that happening at scale either.

But removing CO2 at source is not particularly expensive, can be scaled, and is, I think, in practice a necessity if we want to stabilise below 2 degrees. It is usually not a negative emission though, with some exceptions (e.g. paper mills, or biofueled power plants).

But you do not need negative emissions to stabilise climate at some level, all you need is to reduce your CO2 emissions to zero, then oceans and vegetation take care of the rest. But getting there is hard, and for certain sources it might actually be economical to use direct capture to compensate instead, even at immense cost.

Once we have achieved this, we can start looking more carefully at short lived forcers. Doing it the other way around will only make things worse, which was my point.

Glen Peters on GWP*:

And Zeke Hausfather on a related note:

This is somewhat related to an issue with GWP* that Joeri Rogelj was pointing out:

https://iopscience.iop.org/article/10.1088/1748-9326/ab4928

Ben,

Yes, I did see those. They’re good points. As Thorsten was suggesting, though, this probably illustrates why it’s better to make these kind of arguments, than to use simple metrics. I think GWP* does a good job of estimating warming due to emission of short-lived pollutants, and if one wants to simply determine how to limit warming it might be fine. However, there are other reasons for reducing emissions and it’s probably best to just be explicit about these reasons, rather than rely on some metric.

I must admit that I don’t fully how you can’t use GWP* in a way that accounts for grandfathering, and I’m not quite sure how GWP avoids this. Unless I’m missing something, to avoid this you need to consider past emissions as well as future.

Well, you can use GWP* in a way that avoids grandfathering, like the zero-baseline approach suggested in Rogelj.

But that would create a huge initial ‘bill’ for methane emitting countries/industries, so, unsurprisingly, no-one is suggesting it. Or you could cut carbon budgets for high methane-emitters, to account for the baseline-shift, which would have a similar impact.

i.e. you can apply GWP* in a way that isn’t ridiculously inequitable, but it seems noteworthy that this is not in fact generally what is being done.

Ben,

Indeed, but I don’t get how GWP avoids this. Don’t you have the same problem if you were to use GWP?

Okay, I guess GWP does take into account how the emissions are now, rather than how they’ve changed recently.

Possibly the simplest way of thinking about it is the difference between methane polluters with linearly decreasing polluting being rewarded for polluting less (vanilla, maximally inequitable GWP*) and just facing a reducing penalty (GWP). Thus, rewarding polluters for high levels of historic methane emission.

In ‘less-inequitable’ versions of GWP* accounting, the methane polluters effectively get charged an extra upfront lump-sum, so their total amount they pay is zero. The real cost is that they only get all their money back after their emissions have gone to zero, so they are paying interest on this carbon debt until it leaves the atmosphere.

Ben,

Yes, I see what you mean. I do think the latter option is quite sensible, and probably fair. Possibly hard to implement.

You could argue that we’re not expecting CO2 emitters to reverse any of their past warming, so a pricing mechanism that takes into account that methane emitters who are reducing their emissions are actually reversing some past warming makes some sense to me.

My issue with all these trading systems is that they pretend to be objective, but they are really just different value-based choices that make different trade-offs. As a scientist I can design any metric that fits any set of values you want. But we should really be discussing those values instead of the metrics.

Personally, my greatest concern is that we need to stabilise Earth’s climate and limit end-of-century warming; we need to reduce the current path that we are on towards 3 degrees. If you allow trading CO2 reductions with something related to short lived pollutants, then you are moving further away from that goal since you allow more CO2 to accumulate and hence increase that end-of-century warming. It is that simple. All we do is “kick the can down the road”, “piss our pants to stay warm”, “delay the inevitable”, I think you get it. Therefore, based on my personal values any such trading metric is a diversion away from eliminating fossil fuel emissions, and hence stopping global warming.

Another person might think that warming rates within the next decades is more important to them, and then would focus on short lived forcings. A often promoted selling point is that it gives a quick effect. Based on a set of values that I do not personally sympathise with, this could be a valid point.

In either case the two-basket approach is a more useful starting point than any version of GWP.

I think it is excessively generous to say that someone that stops discharging sewage into a river is ‘reversing past river pollution’, even if their stink eventually washes downstream. This is related to the ‘grandfathering’ idea: “well, I’ve been dumping in the river for all my days, so I shouldn’t have to pay to continue”.

GWP* is also being misused to claim that as long as the stink stays constant, their informal toilet arrangements aren’t contributing to the problem. Flow pollution: it is still pollution.

But the important question is not ‘who is to blame?’. It is how to make people stop polluting the river. The solution should be equitable, i.e. not put too much burden on poor people.

Thorsten,

Yes, I completely agree. My personal view is that we should be mostly concerned about limiting long-term warming. Hence, my concern that a focus on limiting emissions of short-term pollutants could mean emitting more CO2 and, hence, leading to more long-term warming. I would even argue that we should maybe be willing to risk crossing some of warming thresholds temporarily (because we’ve ended up emitting more short-term pollutants that will eventually decay) than risking crossing them permanently (by emitting more CO2 that will persist).

On the other hand, I think people can have different views and can argue in favour of focussing on short-term warming rates. Additionally, there may other perfectly valid arguments for focussing on short-lived pollutants (air quality, for example).

However, I do think that suggesting that there is some objective way to make these assessments is wrong. As you say, all of these decisions include value-based choices and it’s wrong to suggest otherwise. It would, in my view, be much better to be completely open about the various details than to use simple metrics that probably obscure them.

Ben,

Fair point, I hadn’t though of it this way. However, it doesn’t seem entirely fair. In a sense we have two types of pollutants, short-lived and long-lived. Those emitting short-lived pollutants can essentially reverse the direct impact of their emissions (warming) but those who are emitting the long-lived pollutant can only really stop increasing the warming impact of their emissions. It’s – I guess – true that stopping emitting a short-lived pollutant doesn’t completely reverse the impact of the warming, but it does reverse the warming.

If we consider your analogy, then one group cannot do anything about the pollution they’ve already emitted and we seem to let them get away with that, while – in a sense – expecting another group to do something that will essentially reverse some of what they’ve done without giving them any credit for doing so (I may have expressed that poorly).

Indeed, I agree. I’m certainly not suggesting that GWP* isn’t being used in this way and that this isn’t a problem. I just happen to think that if a metric is intended to be used to estimate the warming impact of emissions, then it’s better to use one that does this well, than one that does this poorly.

Absolutely, but aren’t there also potential equity issues with using GWP? For example, in some cases over-estimating the impact of some agricultural emissions.

I suspect we agree more than we disagree 🙂

Well, I think it would be just if the most culpable countries had to make amends for their cumulative historical emissions of CO2, but these countries (this group very much overlaps with those with high current methane emissions) wouldn’t like that very much either.

So Paris emphasises ‘equity’ rather than justice, a sort of blank-slate approach ignoring past emissions, for what are really pragmatic reasons. I think developing countries are already getting a pretty rough deal…

There are all kinds of horrible equity and justice issues no matter what metric you use; I get the impression though that switching to GWP* (I think GWP100 is somehow the default) would re-open questions that were somewhat ‘settled’.

Yes, I don’t think we are really disagreeing that much, it is mostly a question of rhetorical emphasis. The physics is the easy bit. The politics of who does what when is difficult.

Ben,

Agreed.

Possibly. That’s partly why I tend to think it would be good if we could simply be open about the details, but maybe that’s unrealistic and that when you get to politics, simple metrics are easier and more effective.

Indeed, I agree. I do realise that my bias is to focus on the former, rather than the latter. I guess my preference would be when some information is used to justify some policy, that the information (or what it implies) is broadly correct, but that may be a little naive.

ATTP: I think you generally do a good job, actually, of being willing to discuss stuff other than physics. And putting up with commentators with strong opinions who continually want to discuss something slightly orthogonal to the post.

Ben,

Thanks, that’s what I’m trying to do 🙂

“Paris emphasises ‘equity’ rather than justice, a sort of blank-slate approach ignoring past emissions, for what are really pragmatic reasons. I think developing countries are already getting a pretty rough deal…” it’s just real politik. pure power politics at a fundamental level. If push comes to shove, the developed nations can just grind the less-developed nations under their heel. If developed nations try something just, the nationalists in those countries pitch a fit. I might have a slightly skewed view on that as a person from the US of A. I read that 45% of the US population does not yet accept that human activity is the primary driver of global warming. And that 45% is motivated and votes. It looks like a bit of a mess from where I sit.

https://www.vice.com/en/article/qjbd9m/vice-guardian-poll-americans-climate-change-man-made-climate-crimes

It’s not too late. We probably have plenty of time to sort this out. We need almost zero time if an all-powerful deity steps up and makes good on the great flood promise that the Noah era flood was a one-off as many people in the US believe will happen. I guess that could happen. I don’t think I can rule it out, but it would be a real eye-opener for me. As Ken and Ben seem to be agreeing: the politics of who does what when is difficult.

Cheers

Mike

Thorsten: I guess my view is that reducing emissions of greenhouse gases is important and deserves our attention and work. When you look at the gh gases, a sensible person would recognize that CO2 is the “big dog” in the room. I think a sensible person would also take note that CH4 is a nasty little beast in the room as well.

Is it possible to focus on reducing gh gas emissions and not be forced to select CO2 or CH4? Maybe it’s not an “either or” situation? If you consider gh gas emission reductions of all types, there is some low-hanging fruit that we might want to look at. Why do folks often choose to pick low-hanging fruit when there may be larger, tastier fruit up high on the tree?

Also, if you compare the gh gases in a broad way, you may recognize that warming over certain periods of time is not the only consideration that might warrant consideration. With CH4 emission reductions you also get positive impacts on air pollution, crop yield and public health through ozone reduction. Overall, CH4 punches above its weight in terms of bad impacts, but it’s still not the big dog, it’s just a nasty little beast. If we can walk and chew gum, then I think we can work on CO2 and CH4 reductions at the same time. The work on one must not be seen as a trade-off against the other. We need to reduce gh gases.

Cheers

Mike

Mike, yes it is no problem to address CH4, also, as I said in my initial reply. It just shouldn’t, based on my values, get in the way of CO2 emissions reductions. It does so when there is a trading system in place where you can get rewarded for one or the other, which is why I think there shouldn’t be a trading system such as GWP (any version). It also does so when CH4 is constantly touted as “the great danger”, which takes away peoples attention from the fossil fuel industry (which by the way is a major CH4 emitter) by focusing on food and garbage. In an elegant move, not unlike what the tobacco and weapons lobbies did, it is blamed on individuals behaviour, rather than the systemic issue climate mitigation really is.

thank you for the explanation. Does the carbon tax approach settle the problems that arise with a trading system?

For what it is worth, there is actually an existing trading scheme, it is just that the price is $0 on things like CH4 or land use CO2 or general ecosystem destruction.

People making these tradeoffs is already a problem (‘clean’ gas, biofuel). Tradeoffs go in both directions…

My (and I believe this is also reflected in the latest IPCC report ch7) take is the metric used or IF any metric should be used should be determinen by the context. If I understand things correctly, the climate scenarios don’t use any carbon dioxide equivalent metric at all, but estimate the impact of the various gases individually taking into consideration the process of how they break down etc.

Using the GWP metrics in the context of climate credits or trading schemes also seems dubious if you want to have the same effect on the climate.

Most people will encounter the results of the metrics in the shape of LCAs. And then in next step in meta-analysis of LCA:s and in scenarious summing up the results of LCA e.g. for diets etc. All these steps perverts the results. The GWP-100 is also particularly harmful when you compare different ways of breeding cattle. According to LCA:s based on GWP-100 pastoralists’ cattle emitt 5-10 times (I have even seen figures reaching 30 times) more per unit of milk or meat than cattle in developed countries in intensive systems. This is almost totally a methane effect. pastoralists’ cows will normally emit almost no CO2 and hardly any nitrous oxide. In intensive farming, methane’s share of (GWP-100) emissions goes down to around 50 % and carbon dioxide emission and nitrous oxide emissions increase a lot compared to pastoralist management. With GWP-100 one get the result that intensive management is superior.

The number of animals in pastoralist systems are determined by the area of grassland and those have hardly expanded at all apart from in Latin America (the cattle expansion in LA is a special story and should not be mixed up with traditional livestock) and as they don’t feed cattle and don’t use fertilizers or irrigation any given land have the same numbers today as they had 100 years ago. The methane from those cattle contributes no more to global warming, but according to LCA:s each year they increase emissions in a horrific rate.

Of course, you could cull them and lower temperatures, but that is also not reflected by GWP-100. If you cull half of the herd, according to GWP-100 the remainder still give a substantial contribution to global warming year by year, but the real effect is that the temperature will decrease.

🧵👇

Rust,

Thanks, that’s a really good thread.

ATTP/Willard A small favor.

I have been working on maths problems for many years with a special interest [fixation] on Fermat’s theorem.

–

That is no two positive integers to a power of 3 or greater can ever form a third integer to that same power.

–

The answer, which would fit in the margins of a book as Fermat proposed is as follows

“The sum of two cubes is 2 identical cubes plus a gap

When the sum is a cube and the gap is a cube the first cube has to be two identical cubes

Yet two identical cubes can never be a cube.

Therefore the first cube can never be a cube if the gap and sum are cubes.

Only a non cube can add to a cube to make a cube

This proof applies to all powers 3 or greater.”

–

I hope you do not mind me tucking it into your blog.

If it proves reasonable with no flaws please let me know.

If there is a flaw I would be pleased to have it pointed out by the mathematicians here.

angech,

I’m not sure if you’re asking a question, or just pointing something out.

Asking a favour..

Pointing out a simple way of solving a complex problem.

Asking the maths people here to confirm if the answer is valid.

Or to consign it to all the other failed attempts

Many thanks

angech

angech,

I’m not sure why you think that:

Why “2 identical cubes”?

Ah .

All cubes go up by a gap of 6s plus 1

1,8 27, 64 etc

7 6+1, 19 3×6. +1, 37 6×6 +1 etc

When you have 2 cubes , say 8 pls 27 the cube that they would make, if they could make a cube

(They cannot) , would be 35 I.e. 8 plus 8 plus 19.

This is true for any two cubes you wish to add.

That is the first cube is duplicated twice and adds to the gap between the first and second cube.

Instead of trying to prove the gap is a cube (it is not) or the sum of the two cubes is a cube ( it is not)

You just have to show that the sum of two cubes is always 2 identical cubes and a gap.

–

Having done this, true in every case, the problem lies with the assumption that the first cube was a cube.

The result is the gap plus 2 identical cubes.

But this means that the cube that is supposed to add to the gap (assumed to be a cube)

Is actually a double identical cube .

Being irrational the addition can never give true integer cube.

– I would explain this as a double negative induction.

Neither the true identity of the gap Orr the cube that is supposed to be made by two cubes is ever shown but since they demand a double cube to add to the gap it cannot exist..

The logic seems good but must be wrong as somebody should have suggested this line before.

If not we may have a new maths insight.

Thanks ATTP.

Much appreciated.

I was hoping for a better reaction from people.

But that’s life.

I would really that if any of your regulars can give further input or break the argument down better they might try to do so to help me.

” I was hoping for a better reaction from people.”

The reason you didn’t get a better reaction is that you did not bother to express a mathematical problem/solution in mathematical terns, but in extremely ambiguous natural language.

The first part of your solution is true. Let f(n) = n^3, then if your assertion is correct (which it is), then we can write the sequence as a recursion:

f(n) = f(n-1) + 6g(n) + 1

where g(n) is a function of n to be determined. Substituting f(n-1) = (n-1)^3,

f(n) = n^3 = (n-1)^3 + 6g(x) + 1 = n^3 – 3n^2 +3n -1 + 6g(n) + 1

which implies that

g(n) = (n^2 – n)/2

and the original assertion is better expressed as:

f(n) = f(n-1) + 3(n^2 – n) + 1

Unfortunately, the next step is basically word salad: “When you have 2 cubes , say 8 pls 27 the cube that they would make, if they could make a cube (They cannot) , would be 35 I.e. 8 plus 8 plus 19.”

Unfortunately cubes are formed by multiplying not adding. 35 is (3.27106631)^3 so that is the error.

Trying to put this in more mathematical terms, I think you may be trying to say that

a^3 + b^3 = a^3 + a^3 + (b^3 – a^3)

which is obviously true as it is just adding a^3 onto the left hand side of the equation and then subtracting it again, so it is trivially true, but I don’t see how it helps.

“Instead of trying to prove the gap is a cube (it is not) or the sum of the two cubes is a cube ( it is not)

You just have to show that the sum of two cubes is always 2 identical cubes and a gap.”

that tells you nothing about whether a^3 + b^3 = c^3, the fact that c^3 is not involved in your proof demonstrates that. How is

a^3 + a^3 + (b^3 – a^3) = c^3

any easier to prove than

a^3 + b^3 = c^3

which is where we started from.

“Having done this, true in every case, the problem lies with the assumption that the first cube was a cube.

The result is the gap plus 2 identical cubes.”

Yes, we have established that

a^3 + b^3 = a^3 + a^3 + (b^3 – a^3) = a^3 + b^3 + (a^3 – a^3)

“The result is the gap plus 2 identical cubes.

But this means that the cube that is supposed to add to the gap (assumed to be a cube)

Is actually a double identical cube .”

no,

a^3 + b^3 = c^ 3 = 2a^3 + (b^3 – a^3)

not

a^3 + b^3 = c^ 3 = 2a^3

The cube is not “supposed to add to the gap” it is supposed to equal the whole of the left hand side of the equation

“Being irrational the addition can never give true integer cube. ”

I don’t think “irrational” in mathematics means what you think it means.

Thank you very much for your observations DM.

I have just looked at them but as it is after midnight here I will have a better look tomorrow.

I had the pleasure of trying to convince my 39 year old son about this this weekend.

He has a much higher level of university maths than I but is much more into computer programming these days.

He said the same as you, it just doesn’t feel right.

I do think the argument is right so will try to rework it better both mathematically and verbally.

This really means a lot to me whether right or wrong.

Thanks.

DM

in a^3 + b^3 = c^ 3 = 2a^3 + (b^3 – a^3) [thanks]

c^ 3 = 2a^3 + (b^3 – a^3)

The cube equals the whole left side of the equation.

The left hand side has a double cube in it and a gap.

Since 2a^3 is a cube and c^3 has to be a cube there has to be a gap

(b^3 – a^3) which is a cube

in the expression c^ 3 = 2a^3 + x^3 where x^3= (b^3 – a^3)

If this did not exist we cannot have a^3 + b^3 = c^ 3 which is a surrogate form of this equation.

But if we allow x^3= (b^3 – a^3) , even just the once

the 2 a^3 cannot be a cube [ no doubled cube is a cube of an integer]

Hence whether x3 = b^3-a^3 or not

i.e the gap, whatever it is. cannot be a cube.

The beauty of this explanation is that we do not ever have to know the gap to prove that the gap cannot form a cube with a doubled cube.

Any good?

Sorry,

I do not want to run around in circles.

These comments are obviously wrong.

“i.e the gap, whatever it is. cannot be a cube.

The beauty of this explanation is that we do not ever have to know the gap to prove that the gap cannot form a cube with a doubled cube.”

Try again later with more care.

“Since 2a^3 is a cube”

no, I don’t think that is correct.

2a^3 = [2^(1/3) x a]^3

so for 2a^3 to be a cube of an integer would require 2^(1/3) (i.e. the cube root of 2) to be an integer, but it isn’t.

However I haven’t finished my first coffee for the day, so YMMV.

“(b^3 – a^3) which is a cube”

This is not a cube either – it is a difference of two cubes. Again, I think your difficulties lie in your use of natural language. Try providing a derivation using just maths with no natural language.

“But if we allow x^3= (b^3 – a^3) , even just the once”

No, as I pointed out, that is a mathematical necessity that comes from adding a^3 and subtracting it again and then regrouping two of the terms in brackets. It is just algebra.

“i.e the gap, whatever it is. cannot be a cube.” I can’t see where in your argument that the gap needs to be a cube. A cube c^3 can be written as the sum of a non-cube and another non-cube – this is entirely unsurprising.

The wikipedia page on this is useful:

https://en.wikipedia.org/wiki/Proof_of_Fermat%27s_Last_Theorem_for_specific_exponents

i.e. there are simple elegant proofs of FLT for specific exponents (i.e. cubes) that do use reasoning about even/odd numbers.

Oh dear, there is the afternoon gone… ;o)

“But if we allow x^3= (b^3 – a^3) , even just the once”

I’ve just realised that this is just a restatement of the original FMT for cubes where a is a negative number. I did say I’d not even finished my first coffee (Being an minimal Erdos machine of sorts, I can’t prove theorems without it ;o)

Thanks DM and Ben

Does this work

Can a^3 + b^3 = c^3 where a,b and c are positive numbers [integers] >2

note all numbers can be cubes, rational or irrational, but usually they are not numeric cubes

which is required here and throughout the text.

If the theorem is wrong one of the three components has to be proved wrong, this is useful.

The issue here is that the statement implies that the gap between two cubes could,

by some bizarre freak of maths, turn out to be a cube,

even though any particular event can be shown to be not true.

This makes it very difficult as once you accept something might be true

it colors all the maths that is done.

The gap goes up by the triangular series x6 plus 1 for each unit increase.

One way to disprove it would be show that no gap can be a 1 plus 6[triangular series] +1 accumulation. Not worked that out yet.

The other is a bit like the two guards problem where one always tells the truth and the other lies,

complicated by the fact that there are 3 “guards” , not two but made easier because there is only one on one side of the equation.

The statement that every cube can be composed of 2 identical cubes and a gap is essential.

One has to note that a cube can also be made up by 2 identical cubes but in that case,

as DM notes 2a^3 can be a cube, say 8,

but the a component ,4, does not have a cube root.

It works in reverse as well in that 1/2 a^3 cannot be a cube either.

Using real numbers

one can see that 8 plus 27 = 35 where c^3 is not a cube, a^3 and b^3 are cubes

8 + 19 = 27 where b^3 is now the gap and not a cube, a^3 and c^3 are cubes

* the third possibility is

a^3 + b^3 = c^3 where a^3 is not a cube, yet b^3 and c^3 are cubes.

note a^3 is than 50% of c^3/2

[one of the first two cubes being larger than the other]

In 3 the equation can be rewritten as c^3 = 2a^3 +[b^3-a^3]

but in doing so the c^3 could be 27, 35, or some other unspecified cube

The b^3 can be 19, 27 or unspecified but a cube

The a^3 can be 8, 8, or some unspecified value that is a cube, theorem proved,

or not a cube theorem disproved.

When b^3 and c^3 are cubes what is the value of a^3?

All two cube additions require an initial cube, here 8

and another cube say 64 [two gaps as 27 between]

the gaps are 19 to 27 and 37 to 64. The total gap is 56

If the gap, 56, was a cube, then 8 plus 56 = 64

a^3 + b^3 = c^3 where a,b and c are positive numbers [integers] >2

If 8 and 64 are added =72 if the 72 was a cube, case proven.

Now we have the scenario*

1. 8 plus 56 = 64 so 8+64 = 72

Using the idea 2 identical cubes plus a gap = a cube so

8+8 +56 = a cube

We know the c^3 may not be a cube and 56 may not be a cube

but assume that they might be

Now 8+8 [16] cannot have a numerical cube

So whether the other two components, which are the correct components, are cubes or not

The equation as it is shows that as 2a^3 is not a cube.

yet the others have to be

two integer cubes can never make a third integer cube

Hence in its other form it cannot be right either.

“The issue here is that the statement implies that the gap between two cubes could,

by some bizarre freak of maths, turn out to be a cube,”

Unfortunately I have already shown that showing the difference between two cubes could also be a cube is just a restatement of Fermat’s last theorem (in the case of cubes)

a^3 + b^3 = c^3

but if we let d = -a, then this is equivalent to

-d^3 + b^3 = c^3

rearranging

b^3 – d^3 = c^3

so any proof of FMT based on the idea that a difference between two cubes would be circular argument.

Sorry Angech, but you have just added more ambiguous natural language and no additional maths AFAICS, so I can’t really help any further. Note you are also using some mathematical terms incorrectly (e.g. rational/irrational, triangular series etc.) and some ambiguous ones, e.g. “numeric cubes” (is there any other sort?)

” The statement that every cube can be composed of 2 identical cubes and a gap is essential.”

This is incorrect, or at least ambiguous, it should be ” The statement that every cube can be composed of 2 identical cubes and a gap between two cubes is essential.”, However this is easily shown to be true and doesn’t actually make any progress at all. It is just saying that a^3 + b^3 = a^3 + b^3 + a^3 – a^3, where a^3 has been added and subtracted again. I don’t see how that is useful. *why* is it essential?

“as DM notes 2a^3 can be a cube, say 8,”

No, I said the opposite, in fact I gave a proof that 2a^3 can never be a cube.

If it is something you care about, why not take a first-year uni-level course on elementary proofs in basic number theory? Maybe something like the MIT opencourseware “theory of numbers” course.

Then you would be able to have some more productive conversations.

Works for climate science too…

DM you have helped me a lot.

It is appreciated even more because you did not have to go out of your way to do so.

And had every right not to do so.

Maths, statistics and probability and using them properly mean a great deal to me.

Also of course to you, ATTP and a lot of others here.

The circularity you refer to is why this problem has proved virtually insoluble to many mathematicians in the past.

I think it is resolvable and is resolved ( above).

By the approach of proving that the sum 2 cubes can make

Is always replicated by sum of 2 identical numerical cubes and a gap.

When you allow the two identical cubes equal the smaller cube in the formula

You enable the gap and the sum to be cubes

Enjoy the coffee.

Ben McMillan says:

If it is something you care about, why not take a first-year uni-level course on elementary proofs in basic number theory? Maybe something like the MIT opencourseware “theory of numbers” course.

Thanks Ben. All suggestions welcome.

It works like this I guess.

I have , I feel, an adequate grasp of maths to understand most moderate mathematical principles.

The stuff I need is a lot more complex so that I can work through things like Feynman’s lectures and understand the harder things.

Working on this problem has led to some insights on the complexity of going up a dimensional level or two like in string theory.

I would need a 5 year Uni course to get to that level, if I had the sort of brain that could work that level of complexity.

The proof I outlined above works on a double negative induction.

The three components of the problem involve three potential cubes.

When people approach it the gap between the first and final cubes needs to be a cube but it cannot be. The cube that the two cubes make needs to be a cube but cannot be.

The first cube is always presumed to be a cube and so is overlooked in the problem solving.

By using the fact that the second cube, which is actually not a cube and the third cube , which can never be a cube can however be treated as cubes we come to a sum [assumed to be a cube, made with a gap, and a doubled cube of the first cube that we know equals the supposed cube.

We know the gap goes with a cube to make a cube.

Not the same cube as the sum, but it does have to go with a cube.

So now we have the gap making a cube -correct.

We have the doubled cube making a cube with the gap. It has to.

Their sum is a cube, just not the one everyone expects.

So the doubled cube also has to be a cube.

Which it cannot be.

DM “I gave a proof that 2a^3 can never be a cube.” [where a is an integer]

This means of course that a^3 =2.a^3 to fit the result.

Which is impossible.

This makes this one of the elementary proofs.

“By the approach of proving that the sum 2 cubes can make

Is always replicated by sum of 2 identical numerical cubes and a gap.”

this is straight-forward algebra

“When you allow the two identical cubes equal the smaller cube in the formula

You enable the gap and the sum to be cubes”

Again because you don’t write that in algebraic form, but if you mean the 2a^3, we know that is not equal to *any* cube, so that is obviously a non-starter.

BTW I am DM, posting from an account I don’t normally use

“I have , I feel, an adequate grasp of maths to understand most moderate mathematical principles.

The stuff I need is a lot more complex so that I can work through things like Feynman’s lectures and understand the harder things.”

No, the thing you need is very basic, the ability to express your ideas algebraically. If you want to prove something, you need to be able to express yourself without any ambiguity. This is very difficult to do in natural language, which is why mathematicians use algebraic notation.

Feynman was very good at communicating ideas in natural language, but I suspect whenever he wanted to prove something he used mathematical notation (or diagrams representing mathematical notation).

I rather enjoyed Kevin Huston’s “How to think like a mathematician” (although I didn’t find time to go through the exercises). If you are interested in proving things, it is a good read.

http://www1.maths.leeds.ac.uk/~khouston/httlam.html

“We know the gap goes with a cube to make a cube.”

No, this is incorrect. 2a^3 is not a cube for any a. The correct statement is

“We know the gap goes with a non-cube to make a cube”

I made a few posts as my alter-ego, but they are in moderation, so hopefully they should appear at some point.

“This means of course that a^3 =2.a^3 to fit the result.

Which is impossible.”

unless a=1

“When you allow the two identical cubes to equal the smaller cube in the formula

You enable the gap and the sum to be cubes”

Again because you don’t write that in algebraic form, but if you mean the 2a^3, we know that is not equal to *any* cube, so that is obviously a non-starter.

That is part of the reason for the impossibility of the equation being right.

8+8 +19 = 35 where 35 has to be the cube of 8 and 27

[8+8] [as a cube] +19 as a gap =35 as a cube

8 as a cube +19 =27 as a cube

Both equations are right as far as the sum is or would be a cube in both cases whether 19 is a cube or not.

This means 2a^3 and a^3 can make cubes with a cube [19 if 19 were a cube] if it is possible.

We know that 2x 8 is not a cube [of an integer] [ the 2a^3, we know that is not equal to *any* cube,] as you say.

So we could have 2.a^3 +19 = a cube and a^3 +19 = a cube where 19 is also a cube.

since 2a.a^3 cannot be correct for a numerical cube root

19, the putative cube, can only make a cube with a non cube.

2.a^3 +19 = a cube is correct.

2.a^3 is not a cube

so 19, or whatever gap we choose, can only form a cube with a non cube when it is a cube.

This is why no matter what form the gap takes it can only be a cube when the first cube is not a cube.

Which of course makes a lot of sense but was otherwise hard to prove.

“That is part of the reason for the impossibility of the equation being right.”

No angech, the equation does not require that 2a^3 is a cube, just that (2a^3 + b^3 – a^3) is a cube, but that is merely restating the original equation a^3 + b^3 = c^3.

Thanks DM .

I will leave it there for the moment unless anyone else cares to consider it.

Thanks for the maths book suggestion,

taking the interest in the problem, and have a great Xmas.

” I will leave it there for the moment unless anyone else cares to consider it.”

You need to use mathematical notation. You won’t see your error while you use ambiguous natural language.

DM

I have added a second and easier way to explain the reason why two identical cubes or indeed any two identical powers >3 prevent formation of an integer solution to the sum of any 2 integer cubes or powers >3 to a new cube or power.

Since identical cubes can be found in any number all integer cubes>1 can be made up of 2 identical cubes and an integer gap.

Any two identical cubes form a cube with a non integer cube root unless they form a cube with an integer root.

But if they form a cube with an integer root then the identical cubes do not have cube roots that are integers.

This prevents the sum if it is an integer cube root from ever having two integer cube roots cubes making that sum.

Another simple way to prove this,

Both identical cubes have to attach onto a single gap/cube which always needs two different sized cubes to attach to.

The second identical cube does not have a space on the gap it can attach on to hence cannot make a third cube.*

QED

This works not only for cubes but any two positive numbers raised to the same power greater than 2.

a^3 +b^3 /=c^3

b^3 =a^3 + x

a^3 + a^3 + x /=c^3

as x = gap and all x have a smaller cube below to add onto and a larger cube to add onto above.

The cube above cannot be identical to the cube below as a gap always has to have a different sized cube on either side of it.

*a single gap/cube always needs two different sized cubes to attach to.

This is easily proven by a simple formula.

Would this suffice?

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Apparently, GWP* is a model, not a metric…

https://iopscience.iop.org/article/10.1088/1748-9326/ac5930

Ben,

Thanks, I’ll have to look at the paper. I can maybe see why they would regard it as not really being a metric, but it would still seem possible to use it in the place of GWP.