Prouhet's Thue-Morse based solution turns out to be easy enough to prove; despite which I managed to explore a number of culs-de-sac before succeeding with the spoiler below, which does not appear available anywhere online that I could discover. Spot the first typo ... WFL _ _ _ _ _ _ _ _ _ _ _ _ _ _ Note the webpages ignore (g = 0)-th powers: with convention 0^0 = 1 , an `ideal' solution satisfies the same number k+1 of equations as the number n of power terms on each side. Prouhet's Thue-Morse based solution (non-ideal with n = 2^k ) equates to the following. Define s(i) == sum of binary digits of i , f(g, h) == \sum_{0 <= i < 2^h} (-1)^{h+s(i)} i^g ; required to prove f(g, h) = 0 for 0 <= g < h . *** Proof via induction on h : trivial for h = 0 ; so suppose true up to h , and let 0 <= g < h+1 . Then f(g, h+1) = (-1)^{h+1} \sum_{0 <= i < 2^{h+1}} (-1)^{h+1+s(i)} i^g via definition, = - \sum_{0 <= i < 2^h} (-1)^{h+s(i)} i^g + \sum_{0 <= i < 2^h} (-1)^{h+s(i)} ( i^g + g 2^h i^{g-1} + ... ) using s(i + 2^h) = 1 + s(i) in range , along with binomial theorem, = ( -f(g, h) + f(g, h) ) + ( g 2^h f(g-1, h) + ... ) = ( 0 ) + ( 0 + ... ) = 0 via inductive hypothesis. QED Incidentally at g = h , the nonzero power sum above evaluates to f(h, h) = h 2^(h-1) f(h-1, h-1) = h(h-1)(h-2) ... 2^( h-1 + h-2 + ... ) = h! 2^(h_C_2) , *** (i.e. factorial times binomial coefficient). Replacing i^g by (i + a)^g , (b i)^g , i_C_g in the definition of f(g, h) yields results analogous to those above. Fred Lunnon On 6/29/17, Christian Boyer <cboyer@club-internet.fr> wrote:
Pentamagic squares are good examples of sets of integers whose sums of nth powers are equal for n = 1, 2, 3, 4, 5 (and n = 0). Tarry was a specialist on multimagic squares.
Christian. www.multimagie.com
-----Message d'origine----- De : math-fun [mailto:math-fun-bounces@mailman.xmission.com] De la part de Hans Havermann Envoyé : mercredi 28 juin 2017 22:50 À : math-fun <math-fun@mailman.xmission.com> Objet : Re: [math-fun] Equal sums of powers
I recently saw in a book an example of two disjoint sets of integers whose sums of nth powers are equal for n = 1, 2, 3, 4, 5.
https://en.wikipedia.org/wiki/Prouhet–Tarry–Escott_problem
Click on the French version of the wiki for more examples.
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