Talk:Euler's sum of powers conjecture

Euler[edit]

my 9th grade algebra teacher has been discussing things lately with me in our free time about different mathematical theories, and he's talked about about Euler, i was wondering if anyone knows anymore contributions to mathematics he's done, maybe some other problems and unsolved theories. — Preceding unsigned comment added by 208.6.238.226 (talk • contribs) 21:28, 14 April 2003 (UTC)[reply]

Why don't you look at the Euler article? — Preceding unsigned comment added by 67.123.40.12 (talk • contribs) 17:24, 21 February 2004 (UTC)[reply]

Lander-Parkin-Selfridge Conjecture[edit]

Does the conjecture not have an extra stipulation about each a_i being relatively prime to each b_j? There are trivial counterexamples if not (set n = m, choose k $\geq$ n+ m arbitrary and b_i = a_i). —Preceding unsigned comment added by 80.192.83.21 (talk) 13:47, 22 February 2008 (UTC)[reply]

No, the numbers are not required to be co-prime. But a's and b's should be different. I will add that requirement. Maxal (talk) 06:08, 23 February 2008 (UTC)[reply]

I'm removing the k>3 condition as it seems unneeded and the Math Games article doesn't use it. If k is 3 or less then you can have at most 1 term on each side and that would violate the condition a ≠ b condition. Revert if I'm misunderstanding something.--RDBury (talk) 21:06, 15 May 2009 (UTC)[reply]

x^5 + y^5 + z^5 = a^5[edit]

Has anyone searched such a solution in this area : 1 <= x, y, z <= 100000 ? I have searched in 40000 <= x, y, z <= 100000 , and 1 <= x <= 10000 , y, z <= 100000 , no solution . So I wonder whether to continue it or not. --Shenlixing (talk) 22:49, 17 December 2015 (UTC)[reply]

Of the examples listed, only two are actually the sums of two positive fifth powers and another is the sum of two positive fifth powers minus another. Can anyone here see how this is significant?--Myrtonos (talk) 06:10, 9 July 2019 (UTC)[reply]

@Myrtonos: I am afraid that I still have no idea what your question is supposed to mean. --JBL (talk) 12:36, 9 July 2019 (UTC)[reply]

Okay, so there is an intuitive difference between adding two positive numbers and adding a positive and a negative number?--Myrtonos (talk) 13:19, 22 September 2019 (UTC)[reply]

That is an assertion with a question mark at the end, and it has no obvious relationship to the present article. --JBL (talk) 14:23, 22 September 2019 (UTC)[reply]

But it has a relationship to something that I added to the article, that seems intuitive to state, but it was removed without even asking why it should be pointed out.--Myrtonos (talk) 08:01, 23 September 2019 (UTC)[reply]

You should spend a few minutes thinking about what information would be necessary for someone other than you to understand what you are trying to say here. Then you should include all that information in a single comment, with a clear indication of what edit you would like to make and why. Then it would be possible that someone might understand whatever it is that you're trying to say. --JBL (talk) 11:54, 23 September 2019 (UTC)[reply]

I have, and separating the sums of only positive numbers from sums with negative numbers would help. When a layperson thinks of adding, they think of adding positive numbers.--Myrtonos (talk) 09:31, 26 October 2019 (UTC)[reply]

I still don't see why it shouldn't be mentioned. --Myrtonos (talk) 12:22, 7 September 2020 (UTC)[reply]

I found another example:[edit]

a=54, b=168, c=220, d=266, e=288

where

a5 =459165024, b5=133827821568, c5=515363200000, d5=1331705468576, e5=1981355655168 where a5 + b5 + c5 + d5= e5

— Preceding unsigned comment added by Syhliu (talk • contribs) 19:44, 13 November 2012 (UTC)[reply]

Examples whose GCF is not 1 can be reduced and are thus trivial. Georgia guy (talk) 23:57, 13 November 2012 (UTC)[reply]

More precisely, your [a,b,c,d,e] = 2*[27, 84, 110, 133, 144], the latter being given in the text. — MFH:Talk 17:12, 11 October 2018 (UTC)[reply]

The least natural numbers[edit]

n=2--->2 n=3,4--->3 n=5~8--->4 n=9~16--->5 n=17~32--->6 n=33~64--->7 ...... — Preceding unsigned comment added by 140.113.136.219 (talk) 09:05, 14 April 2014 (UTC)[reply]

A relative conjecture[edit]

If $\sum _{i=1}^{n}a_{i}^{k}=b^{k}$ , than k ≤ 2^{n - 1}, is it right? For example, 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, and 5 ≤ 2^{4 - 1} = 8. Besides, if k ≤ 2^{n - 1}, than $\sum _{i=1}^{n}a_{i}^{k}=b^{k}$ has infinitely many solutions, is it right? — Preceding unsigned comment added by 140.115.142.120 (talk) 04:16, 31 October 2014 (UTC)[reply]

That conjecture is open, but as of 2014, there's no official name of it according to any reliable source. Can anyone (not Wikipedia please) reveal a source with the conjecture's name?? Georgia guy (talk) 14:02, 31 October 2014 (UTC)[reply]

Concerning the second part, you have k = 6 < 2^(4-1) = 8, but there is no solution known for k=6, n=4, and the LPS conjecture rather indicates that there can't be a solution with n < 5. Concerning the first part, you just terribly weaken the LPS inequality k <= n + 1 <= 2^(n-1) for all n > 2. — MFH:Talk 17:24, 11 October 2018 (UTC)[reply]

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Counterexamples for k=5[edit]

Hi, The 'counterexamples' section mentions:

"A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known:"

But the second example is not in positive integers, which is in contrast with the beginning of the article:

"It states that for all integers n and k greater than 1, if the sum of n many kth powers of positive integers is itself a kth power" 185.48.129.79 (talk) 23:34, 17 May 2023 (UTC)[reply]