Mengoli's Six-Square Problem
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From: TPiezas on 26 Jul 2010 06:22 Hello all, Pietro Mengoli's Six-Square Problem (1674) asks to find three integers {x,y,z} such that the sum and differences of any two is a square. Euler found a soln as, {x,y,z} = {434657, 420968, 150568} x+y = 925^2 x-y = 117^2 y+z = 756^2 y-z = 520^2 x+z = 765^2 x-z = 533^2 Notice that, (925*117)^2 + (756*520)^2 = (765*533)^2 If we define a bound B as one where all positive terms {x,y,z} are less than B, then Euler's is the smallest known so far. Can you find a smaller one? For more details, see: http://sites.google.com/site/tpiezas/updates07 - Titus
From: TPiezas on 26 Jul 2010 06:28 On Jul 26, 4:22 am, TPiezas <tpie...(a)gmail.com> wrote: > Hello all, > > Pietro Mengoli's Six-Square Problem (1674) asks to find three integers > {x,y,z} such that the sum and differences of any two is a square. > Euler found a soln as, > > {x,y,z} = {434657, 420968, 150568} > > x+y = 925^2 > x-y = 117^2 > y+z = 756^2 > y-z = 520^2 > x+z = 765^2 > x-z = 533^2 > > Notice that, > > (925*117)^2 + (756*520)^2 = (765*533)^2 > > If we define a bound B as one where all positive terms {x,y,z} are > less than B, then Euler's is the smallest known so far. > > Can you find a smaller one? > > For more details, see:http://sites.google.com/site/tpiezas/updates07 > > - Titus To make things harder, the differences of any two must be a NON-ZERO square... - Titus
From: Tim Little on 26 Jul 2010 22:53 On 2010-07-26, TPiezas <tpiezas(a)gmail.com> wrote: > Pietro Mengoli's Six-Square Problem (1674) asks to find three integers > {x,y,z} such that the sum and differences of any two is a square. > Euler found a soln as, > > {x,y,z} = {434657, 420968, 150568} [...] > Can you find a smaller one? By straightforward exhaustive search, there is none. There may be a simple proof that Euler's solution is the smallest but it eludes me. - Tim
From: Bill Taylor on 27 Jul 2010 02:21 > > Pietro Mengoli's Six-Square Problem (1674) asks to find three integers > > {x,y,z} such that the sum and differences of any two is a square. > > Euler found a soln as, > > > {x,y,z} = {434657, 420968, 150568} > By straightforward exhaustive search, there is none. There may be a > simple proof that Euler's solution is the smallest but it eludes me. How common/how many larger solutions do we expect there to be? What are the gross heuristics on this matter. Anyone? -- Brash Bill ** Officer, officer I've been graped! ** Don't you mean raped? ** No, there was a bunch of them.
From: TPiezas on 27 Jul 2010 03:05
On Jul 26, 8:53 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-07-26, TPiezas <tpie...(a)gmail.com> wrote: > > > Pietro Mengoli's Six-Square Problem (1674) asks to find three integers > > {x,y,z} such that the sum and differences of any two is a square. > > Euler found a soln as, > > > {x,y,z} = {434657, 420968, 150568} > [...] > > Can you find a smaller one? > > By straightforward exhaustive search, there is none. There may be a > simple proof that Euler's solution is the smallest but it eludes me. > > - Tim It turns out that Mengoli's six-square problem is equivalent to finding what is called a "face cuboid" (a Euler brick that is one equation shy of a perfect cuboid) and can be expressed in smaller numbers. Let, {2x, 2y, 2z} = {2a^2+b^2+c^2, b^2+c^2, -b^2+c^2} where b < c and the three expressions, {a^2+b^2, a^2+c^2, a^2+b^2+c^2} are squares. (Of course, if b^2+c^2 is a square as well, that would already be a perfect cuboid.) I found 5 solns {a,b,c} < 1000, though I don't know if this is a complete list. Arranged by the variable "a", {104, 153, 672} {117, 520, 756} (Euler's) {448, 264, 975} {495, 264, 952} {840, 448, 495} As {b,c} must have the same parity for unscaled {x,y,z} to be integers, the 2nd yields Euler's soln and has smaller integer {x,y,z} than the 1st. What is the complete list {a,b,c} < 2000? - Titus |