# Question about Fermat's Last Theorem **URL:** \ **Category:** Factual Questions\ **Created:** [March 17, 2014, 10:41pm UTC](https://boards.straightdope.com/t/question-about-fermats-last-theorem/683959 "2014-03-17T22:41:39Z")\ **Posts on this page:** 9\ **Page:** 2
**Author:** ![Itself](https://avatars.discourse-cdn.com/v4/letter/i/d07c76/32.png) [@Itself](https://boards.straightdope.com/u/Itself)\ **Post date:** [March 18, 2014, 3:49am UTC](https://boards.straightdope.com/t/question-about-fermats-last-theorem/683959/21 "2014-03-18T03:49:15Z")
> [@Indistinguishable](#): > > All I’ve found is that Euler’s sum of powers conjecture is open for n \>= 6, but this has to do with whether one can find a sum of _less than n_ many nth powers which is itself an nth power, and thus does not directly answer my question. Oops! Yes, of course. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [March 18, 2014, 5:13am UTC](https://boards.straightdope.com/t/question-about-fermats-last-theorem/683959/22 "2014-03-18T05:13:15Z")
Problems like > [@Just\_Asking\_Questions](#): > > a[sup]3[/sup] + b[sup]3[/sup] + c[sup]3[/sup] = d[sup]3[/sup] are loosely related to [Waring’s Problem](http://en.wikipedia.org/wiki/Waring's_problem): _Can all (sufficiently large) integers be expressed as the sum of n k’th powers?_ For example, it is [conjectured that all integers larger than 7.4 trillion can be expressed as the sum of four non-negative cubes](http://www.ams.org/journals/mcom/2000-69-229/S0025-5718-99-01116-3/). While not as famous as Fermat’s Last Theorem, Waring’s Problem has attracted huge interest; proofs for various cases are associated with the names of some of the greatest mathematicians. ---
**Author:** ![Just\_Asking\_Questions](https://avatars.discourse-cdn.com/v4/letter/j/ba9def/32.png) [@Just\_Asking\_Questions](https://boards.straightdope.com/u/Just_Asking_Questions)\ **Post date:** [March 18, 2014, 5:20am UTC](https://boards.straightdope.com/t/question-about-fermats-last-theorem/683959/23 "2014-03-18T05:20:00Z")
> [@KarlGauss](#): > > And, just for the record, the OP’s question is referred to as [Euler’s Sum of Powers Conjecture](http://en.wikipedia.org/wiki/Euler's_sum_of_powers_conjecture). This is great! This is what I was thinking of! I would have never found that on my own. Interesting that there are solutions for ‘lesser’ integers. The example given for the 5th power: 27[sup]5[/sup] + 84[sup]5[/sup] + 110[sup]5[/sup] + 133[sup]5[/sup] = 144[sup]5[/sup] means there is a solution with only 4 variables instead of 5. So there is no pattern of increasing variables matching increasing exponents like I was imagining. So in the case of FLT, if I am understanding this correctly, is that for a[sup]n[/sup] + b[sup]n [/sup] = c[sup] n [/sup], there is no ‘lesser’ power answer (for n=3, for example), which invalidates the progression I was imagining, which is why FLT is so important, I gather. Neat! ---
**Author:** ![Just\_Asking\_Questions](https://avatars.discourse-cdn.com/v4/letter/j/ba9def/32.png) [@Just\_Asking\_Questions](https://boards.straightdope.com/u/Just_Asking_Questions)\ **Post date:** [March 18, 2014, 5:26am UTC](https://boards.straightdope.com/t/question-about-fermats-last-theorem/683959/24 "2014-03-18T05:26:18Z")
**Itself, KarlGauss, Indestinguishable** , and anyone I’ve missed, thanks for the replies. ---
**Author:** ![j\_sum1](https://avatars.discourse-cdn.com/v4/letter/j/8baadc/32.png) [@j\_sum1](https://boards.straightdope.com/u/j_sum1)\ **Post date:** [March 18, 2014, 5:49am UTC](https://boards.straightdope.com/t/question-about-fermats-last-theorem/683959/25 "2014-03-18T05:49:23Z")
I don’t have it off the top of my head (unsurprisingly). I do recall reading a search on the sum of five fifth powers equalling another fifth power. Two solutions discovered had 0[sup]5[/sup] as a one of the terms. In other words, there are (at least) two known instances where the sum of four fifth powers equals another fifth power. that is, a[sup]5[/sup] + b[sup]5[/sup] + c[sup]5[/sup] + d[sup]5[/sup] = e[sup]5[/sup] That, to me is interesting. When I see such things I always want to know if there is a generalised pattern or some way of finding/predicting further similar incidents. ---
**Author:** ![Just\_Asking\_Questions](https://avatars.discourse-cdn.com/v4/letter/j/ba9def/32.png) [@Just\_Asking\_Questions](https://boards.straightdope.com/u/Just_Asking_Questions)\ **Post date:** [March 18, 2014, 1:35pm UTC](https://boards.straightdope.com/t/question-about-fermats-last-theorem/683959/26 "2014-03-18T13:35:59Z")
> [@j\_sum1](#): > > That, to me is interesting. When I see such things I always want to know if there is a generalised pattern or some way of finding/predicting further similar incidents. That’s what got me started on this. I wanted a pattern, and it seemed that FLT was going against the ‘pattern’. That it, if every increase in exponent required a corresponding increase in number of variables, then it was silly to assume that there would be solutions for ‘number of exponent minus 1’ number of variables. But since there are, as seen in the wikipedia page, the pattern I was looking for does not exist. Now it makes sense why Fermat was trying to prove there were none for exponent 2: that becomes the ‘odd’ case. ---
**Author:** ![Isilder](https://avatars.discourse-cdn.com/v4/letter/i/cc9497/32.png) [@Isilder](https://boards.straightdope.com/u/Isilder)\ **Post date:** [March 19, 2014, 10:42am UTC](https://boards.straightdope.com/t/question-about-fermats-last-theorem/683959/27 "2014-03-19T10:42:08Z")
> [@Just\_Asking\_Questions](#): > > I am not a mathematician. Probably this is an incredibly obvious question, but I have never read anything about it. Likely because I can’t ask the right question to find an explaination. So I turn to the Teeming Millions: > > Fermat’s Last Theorem states that no three positive integers a, b, and c can satisfy the equation a[sup]n[/sup]+ b[sup]n[/sup]=c[sup]n[/sup] for any integer value of n greater than two. > > a[sup]2[/sup] + b[sup]2[/sup]= c[sup]2[/sup] having solutions is known to all and sundry. Fermat’s theorem states there are no solutions for a[sup]3[/sup] + b[sup]3[/sup]= c[sup]3[/sup] etc etc. > > My question is, why isn’t the progression seen as: > > a[sup]3[/sup] + b[sup]3[/sup] + c[sup]3[/sup] = d[sup]3[/sup] advances in solving the unsolved “Fermat” and other conjectures may be useful in real world problems. eg The Fermat conjecture may be paralleling equations (involving squares…) that lead to prime numbers… Primes are useful for cryptography. and other problems… Questions that have easy answers don’t get called “Fermat’s 953246th conjecture”. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [March 19, 2014, 4:32pm UTC](https://boards.straightdope.com/t/question-about-fermats-last-theorem/683959/28 "2014-03-19T16:32:00Z")
> [@Just\_Asking\_Questions](#): > > Now it makes sense why Fermat was trying to prove there were none for exponent 2: that becomes the ‘odd’ case. It’s not that Fermat was _trying_ to prove it; it’s that he (in a flash of insight?) had what he thought _was_ a “truly marvelous proof.” If mathematicians think you have a “truly marvelous proof” of _anything_, they’ll get all excited about it. ---
**Author:** ![Itself](https://avatars.discourse-cdn.com/v4/letter/i/d07c76/32.png) [@Itself](https://boards.straightdope.com/u/Itself)\ **Post date:** [March 20, 2014, 7:06pm UTC](https://boards.straightdope.com/t/question-about-fermats-last-theorem/683959/29 "2014-03-20T19:06:31Z")
> [@Isilder](#): > > advances in solving the unsolved “Fermat” and other conjectures may be useful in real world problems. > > eg The Fermat conjecture may be paralleling equations (involving squares…) that lead to prime numbers… Primes are useful for cryptography. and other problems… Mathematicians, especially theoretical mathematicians working in number theory, are not particularly interested in real-world problems. 🙂 (Also, despite the importance of elliptic curves in cryptography, the math involved the proof of FLT, or more specifically the modularity conjecture, is light-years beyond anything involved in real-world problems, at least for the foreseeable future. It’s not unlikely that there will be some application eventually, but mathematicians aren’t interested in FLT because they’re salivating over crypto applications.) [Previous page](https://boards.straightdope.com/t/question-about-fermats-last-theorem/683959.md?page=1)