From: randallr@ABAC.COM (Randall L. Rathbun) Subject: Computer searches for the Integer Cuboid - an update Date: 20 Dec 99 12:53:42 GMT Newsgroups: sci.math.numberthy Computer searches for the perfect integer cuboid - Recently there has been discussion about computer searches for possible solutions to the Rational Cuboid Problem (D18 in Dr. Richard K. Guy's Book Unsolved Problems in Number Theory, Springer-Verlag, ed. 2). To avoid further useless reduplication of effort, the following information is shared. Integer cuboids occur in 3 types, depending upon which one length of the seven possible ( 3 edges, 3 face diagonals, 1 body diagonal) in this rectangular parallelopiped are irrational. They are body, edge, and face cuboids wherein the body diagonal, an edge, or a face diagonal is irrational respectively. Representative first members of each class ( 3 edges, body diagonal) are the 44,117,240,sqrt(73225) body cuboid, the 124, 957 sqrt(13852800) 3845 edge cuboid and the 672 153 104 697 face cuboid. The face diagonal between the edges 672 and 153 is sqrt(474993). It is a notorious open question as to whether or not a perfect integer cuboid exists with all 7 lengths integer. Can a possible cuboid be found, if just an edge is known? The answer is yes, and the following procedure outlines how such cuboids may be found by computer searching. Procedure to search for a cuboid by its edge - Let d be a proper divisor of N^2, where N is an integer. Further constrain d so that 0 None 2) A^2 + b^2 = S^2 -> a,b,N,S face cuboid 3) A^2 - B^2 = S^2 -> same as a^2-b^2 4) A^2 - b^2 = S^2 -> N, b, square root(A^2-B^2), A edge cuboid 5) a^2 + B^2 = S^2 -> same as A^2+b^2 6) a^2 + b^2 = S^2 -> a,b,N,square root(S^2+N^2) body cuboid 7) a^2 - B^2 = S^2 -> N,a,square root(B^2-A^2),B edge cuboid 8) a^2 - b^2 = S^2 -> S,N,b,A face cuboid In our example of N=44, we find by computer searching that 240^2 + 117^2 = 267^2. This is the #6 relationship above, a^2 + b^2 = S^2, so the integer cuboid found is a body cuboid of edges 240, 117, 44, body diagonal square root (267^2+44^2 or 73225). This is the smallest such body cuboid that does actually exist. The selection of N=44 is a deliberate choice of the smallest side of this cuboid and shows how a cuboid can be obtained from its smallest side. Utilizing a general purpose computer algorithm which checks all 8 relationships, a computer search for 1 < N <= 1,281,000,000 was conducted. This search was done over a cluster of 12 IBM RS-6000 workstations and took almost exactly 5 years of computer time to run. Over 30,000 cuboids in the ratio of body:face:edge = 3:3:2 were found. No perfect integer cuboids were found however. Thus if a perfect integer cuboid exists, its smallest side must exceed 1.281E9 as the computer search was exhaustive over its range. This computer search duplicated an earlier search over the range 1 to 333,750,000 and identical results were obtained, although different algorithms and programming was employed. High multiple precision arithmetic was employed for accuracy so that no solutions were missed. Searching for a cuboid by its body diagonal - Integer cuboids can also be located by considering the space or body diagonal. Call this diagonal D. This diagonal will usually be a product Legendre primes of the form 1 mod 4. Find all possible sums of X^2+x^2 = D^2. (X>x and x>0) Then search for the following relationships among any two pair of X,x and Y,y where we label Y,y as 2nd smaller set of X,x where Y X Y sqrt(D^2-S^2) D imaginary edge cuboid 2) X^2 + y^2 = S^2 -> X y sqrt(D^2-S^2) D imaginary edge cuboid 3) x^2 + y^2 = S^2 -> x y sqrt(D^2-S^2) D edge cuboid 4) Y^2 + x^2 = S^2 -> Y x sqrt(D^2-S^2) D edge cuboid 5) X^2 - Y^2 = S^2 -> S Y x D face cuboid 6) y^2 - x^2 = S^2 -> S x Y D face cuboid 7) X^2 - y^2 = S^2 -> S y x D face cuboid 8) Y^2 - x^2 = S^2 -> S x y D face cuboid example 1): 60 63 sqrt(-3344) 65 example 2): 364 240 sqrt(-56871) 365 example 3): 520 576 sqrt(618849) 1105 example 4): 1800 1443 sqrt(461776) 2405 It should be noted that in face cuboids, if they occur, case 5) and case 6) occurs together, or else case 7) and case 8) does. An example of cases 5,6) is: 680^2 - 672^2 = 104^2 185^2 - 153^2 = 104^2 which gives the 104 153 672 697 face cuboid. An example of cases 7,8) is: 975^2 - 495^2 = 840^2 1073 952^2 - 448^2 = 840^2 which gives the 840 448 495 1073 face cuboid. Again, to illustrate the relationships above, D=1105 was chosen. Then the following 13 X,x pairs were found: X,x = [1104,47] [1100,105] [1092,169] [1073,264] [1071,272] [1020,425] [1001,468] [975,520] [952,561] [943,576] [884,663] [855,700] [817,744] A computer search of relationships 1-8 listed above for this set finds that: 1073^2 - 952^2 = 495^2 and 561^2 - 264^2 = 495^2 this creates the 495 952 264 1105 face cuboid 1073^2 - 975^2 = 448^2 and 520^2 - 264^2 = 448^2 this creates the 448 975 264 1105 face cuboid Pairs of face cuboids sharing a common edge are occasionally found, this is the smallest such occurrance. The computer also finds that: 520^2 + 576^2 = 776^2 this creates the 520 576 sqrt(618849) 1105 edge cuboid. Thus for the body diagonal length of 1105, two face cuboids and one edge cuboid exists. If the body diagonal consists of the product of many Legendre prime facts, typically several edge and face cuboids are found, sharing the same body diagonal. In the example of D=1105 we see 1105 = 5 * 13 * 17 or 3 Legendre primes. The relationships given above for a cuboid and its edges or body diagonal have been successfully programmed as computer algorithms. It should be noted that the edge relationships will locate all 3 types of cuboids. The body diagonal relationships will miss the body cuboid case although it finds all face or edge cuboid solutions for a given body diagonal. Finally while it is my conjecture that the perfect cuboid does not exist, Derrick N. Lehmer has aptly pointed out, "Happiness is just around the corner!" - Randall L. Rathbun