u^6 + v^6 + w^6 + x^6 + y^6 = z^6
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From: Frederick Williams on 27 Jul 2010 16:57 Euler conjectured that u^6 + v^6 + w^6 + x^6 + y^6 = z^6 had no nonnegative integral solutions unless at least four variables were zero. Was he right? -- I can't go on, I'll go on.
From: Gerry Myerson on 27 Jul 2010 19:27 In article <4C4F4842.70B37535(a)tesco.net>, Frederick Williams <frederick.williams2(a)tesco.net> wrote: > Euler conjectured that u^6 + v^6 + w^6 + x^6 + y^6 = z^6 had no > nonnegative integral solutions unless at least four variables were > zero. Was he right? Have a look at http://euler.free.fr/ If I'm reading it correctly, the allegation there is that no one has found 6 6th powers adding to a 6th power, let alone 5. Do a search for "sums of like powers." -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Frederick Williams on 29 Jul 2010 09:32 Gerry Myerson wrote: > > In article <4C4F4842.70B37535(a)tesco.net>, > Frederick Williams <frederick.williams2(a)tesco.net> wrote: > > > Euler conjectured that u^6 + v^6 + w^6 + x^6 + y^6 = z^6 had no > > nonnegative integral solutions unless at least four variables were > > zero. Was he right? > > Have a look at http://euler.free.fr/ Thanks for the link... > If I'm reading it correctly, the allegation there is that > no one has found 6 6th powers adding to a 6th power, > let alone 5. > > Do a search for "sums of like powers." .... and the suggestion. The question arises from a problem in Knuth's AoCP but since it was published in the seventies I wondered if things had moved on. -- I can't go on, I'll go on.
From: TPiezas on 29 Jul 2010 10:29
On Jul 29, 7:32 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > Gerry Myerson wrote: > > > In article <4C4F4842.70B37...(a)tesco.net>, > > Frederick Williams <frederick.willia...(a)tesco.net> wrote: > > > > Euler conjectured that u^6 + v^6 + w^6 + x^6 + y^6 = z^6 had no > > > nonnegative integral solutions unless at least four variables were > > > zero. Was he right? > > > Have a look athttp://euler.free.fr/ > > Thanks for the link... > > > If I'm reading it correctly, the allegation there is that > > no one has found 6 6th powers adding to a 6th power, > > let alone 5. > > > Do a search for "sums of like powers." > > ... and the suggestion. > > The question arises from a problem in Knuth's AoCP but since it was > published in the seventies I wondered if things had moved on. > > -- > I can't go on, I'll go on. Meyrignac formalized the "Lander-Parkin-Selfridge Conjecture" (LPS) as, "The eqn x1^k + x2^k + ... + x_m^k = y1^k + y2^k + ... + y_n^k has a non-trivial solution in the integers for k > 3 if and only if m+n => k." Thus, a^5+b^5+c^5+d^5 = e^5 (a 5.1.4) supposedly has a soln (and in fact it does) but a^5+b^5+c^5 = d^5 (a 5.1.3) supposedly does not. Unfortunately, no one has proved LPS for any exponent. Both (6.1.5) and (6.1.6) are assumed to be solvable, but none have yet been found and, for the latter, any soln must have a sum > 500,000^6. For higher powers, they have found a (8.1.8), (8.4.4), (8.3.5), but not yet a (8.2.6) or (8.1.7). With computers getting faster every year, an efficient algorithm, and a distributed effort, it might be feasible to tackle again (6.1.6) later this decade (since Meyrignac has stopped the search for 6.1.6 a few years ago). - Titus |