Mengoli's Six-Square Problem
in [Math]
|
Prev: Topology Puzzle
Next: turning probability combinations and permutations into geometrical concepts #4.20 & #236 Correcting Math & Atom Totality
From: Raymond Manzoni on 27 Jul 2010 14:14 TPiezas a �crit : > 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 {104,153,672} {117,520,756} {208,306,1344} {234,1040,1512} {448,264,975} {495,264,952} {840,448,495} {896,528,1950} {990,528,1904} {1680,896,990} {1881,1092,1540} {a,b,c} < 10000 {104,153,672} {117,520,756} {208,306,1344} {234,1040,1512} {252,2261,2640} {312,459,2016} {333,644,2040} {351,1560,2268} {399,468,4180} {416,612,2688} {448,264,975} {468,2080,3024} {495,264,952} {504,4522,5280} {520,765,3360} {585,2600,3780} {624,918,4032} {666,1288,4080} {702,3120,4536} {728,1071,4704} {756,533,3360} {756,6783,7920} {798,936,8360} {819,3640,5292} {832,1224,5376} {840,448,495} {896,528,1950} {936,1377,6048} {936,4160,6048} {990,528,1904} {999,1932,6120} {1040,1530,6720} {1053,4680,6804} {1144,1683,7392} {1170,5200,7560} {1248,1836,8064} {1276,357,6960} {1287,5720,8316} {1332,2576,8160} {1344,792,2925} {1352,1989,8736} {1404,6240,9072} {1456,2142,9408} {1485,792,2856} {1512,1066,6720} {1521,6760,9828} {1680,819,3740} {1680,896,990} {1680,1925,2052} {1680,3404,4653} {1771,1428,2640} {1792,1056,3900} {1827,1564,8736} {1881,1092,1540} {1932,2880,4301} {1980,1056,3808} {2240,1320,4875} {2352,5236,7011} {2475,1320,4760} {2520,1344,1485} {2640,2275,2772} {2688,1584,5850} {2970,1584,5712} {3124,4557,9840} {3136,1848,6825} {3360,1638,7480} {3360,1792,1980} {3360,3850,4104} {3360,6808,9306} {3465,1848,6664} {3542,2856,5280} {3584,2112,7800} {3627,840,1364} {3762,2184,3080} {3864,5760,8602} {3960,2112,7616} {4032,2376,8775} {4032,6601,8976} {4200,2240,2475} {4389,2548,8352} {4455,2376,8568} {4480,2640,9750} {4950,2640,9520} {5040,2688,2970} {5040,5775,6156} {5280,4550,5544} {5313,4284,7920} {5605,2772,7800} {5643,3276,4620} {5852,861,6864} {5880,3136,3465} {6048,1665,4264} {6536,3927,5952} {6720,3584,3960} {6720,7700,8208} {7254,1680,2728} {7524,4368,6160} {7560,4032,4455} {7920,6825,8316} {8400,4480,4950} {9240,4928,5445} {9405,5460,7700} and so on (evaluation is fast...) Hoping it helped, Raymond
From: TPiezas on 27 Jul 2010 16:40
On Jul 27, 12:14 pm, Raymond Manzoni <raym...(a)free.fr> wrote: > TPiezas a écrit : > > > > > > > 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 > > {104,153,672} > {117,520,756} > {208,306,1344} > {234,1040,1512} > {448,264,975} > {495,264,952} > {840,448,495} > {896,528,1950} > {990,528,1904} > {1680,896,990} > {1881,1092,1540} > > {a,b,c} < 10000 > > {104,153,672} > {117,520,756} > {208,306,1344} > {234,1040,1512} > {252,2261,2640} > {312,459,2016} > {333,644,2040} > {351,1560,2268} > {399,468,4180} > {416,612,2688} > {448,264,975} > {468,2080,3024} > {495,264,952} > {504,4522,5280} > {520,765,3360} > {585,2600,3780} > {624,918,4032} > {666,1288,4080} > {702,3120,4536} > {728,1071,4704} > {756,533,3360} > {756,6783,7920} > {798,936,8360} > {819,3640,5292} > {832,1224,5376} > {840,448,495} > {896,528,1950} > {936,1377,6048} > {936,4160,6048} > {990,528,1904} > {999,1932,6120} > {1040,1530,6720} > {1053,4680,6804} > {1144,1683,7392} > {1170,5200,7560} > {1248,1836,8064} > {1276,357,6960} > {1287,5720,8316} > {1332,2576,8160} > {1344,792,2925} > {1352,1989,8736} > {1404,6240,9072} > {1456,2142,9408} > {1485,792,2856} > {1512,1066,6720} > {1521,6760,9828} > {1680,819,3740} > {1680,896,990} > {1680,1925,2052} > {1680,3404,4653} > {1771,1428,2640} > {1792,1056,3900} > {1827,1564,8736} > {1881,1092,1540} > {1932,2880,4301} > {1980,1056,3808} > {2240,1320,4875} > {2352,5236,7011} > {2475,1320,4760} > {2520,1344,1485} > {2640,2275,2772} > {2688,1584,5850} > {2970,1584,5712} > {3124,4557,9840} > {3136,1848,6825} > {3360,1638,7480} > {3360,1792,1980} > {3360,3850,4104} > {3360,6808,9306} > {3465,1848,6664} > {3542,2856,5280} > {3584,2112,7800} > {3627,840,1364} > {3762,2184,3080} > {3864,5760,8602} > {3960,2112,7616} > {4032,2376,8775} > {4032,6601,8976} > {4200,2240,2475} > {4389,2548,8352} > {4455,2376,8568} > {4480,2640,9750} > {4950,2640,9520} > {5040,2688,2970} > {5040,5775,6156} > {5280,4550,5544} > {5313,4284,7920} > {5605,2772,7800} > {5643,3276,4620} > {5852,861,6864} > {5880,3136,3465} > {6048,1665,4264} > {6536,3927,5952} > {6720,3584,3960} > {6720,7700,8208} > {7254,1680,2728} > {7524,4368,6160} > {7560,4032,4455} > {7920,6825,8316} > {8400,4480,4950} > {9240,4928,5445} > {9405,5460,7700} > > and so on (evaluation is fast...) > > Hoping it helped, > Raymond- Hide quoted text - > > - Show quoted text - Thanks, Raymond! (I also got a private email from Jarek Wroblewski who gave me a list with a < 100000!) Anyway, to recap, among other things, we wish to know if Euler's soln {x,y,z} = {434657, 420968, 150568} with x > y > z has the smallest positive integer x that solves Mengoli's six-square problem, Since x = (2a^2+b^2+c^2)/2 where, a^2+b^2, a^2+c^2, a^2+b^2+c^2 are all squares (and which describes a "face cuboid"), then the question is reduced to merely enumerating all face cuboids with sides {a,b,c} < 1000. (If it has a side > 1000, then x obviously will be > 500,000.) There are only 5, {104, 153, 672} {117, 520, 756} (Euler's) {448, 264, 975} {495, 264, 952} {840, 448, 495} which is the same list I got from Rathbun (see http://www.math.niu.edu/~rusin/known-math/99/cuboid). Since none of the others yields smaller {x,y,z}, then Euler's is the smallest soln to Mengoli's six-square problem. (QED) In fact, it turns out there are only 3 {x,y,z} with all terms less than a million, {434657, 420968, 150568} {733025, 488000, 418304} {993250, 949986, 856350} - Titus proving that Euler's soln to Mengoli's six-square problem can be reduced to enumerating all "face cuboids" with sides {a,b,c} < 1000 |