Karl Scherer asked: Nov. 1996
Find the integer solutions of
x1*x2*x3*x4*x5*x6*x7*x8 = x1^2 + x2^2 + ... + x8^2
Can you find a solution with x7 = 7 ?
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This is a special case of the Hurwitz equation:
x1^2 + x2^2 + ... + xn^2 = A x1 x2 ... xn.
Only for 1<=A<=n there are integral solutions.
For any n there is a finite set of solutions
which generate all others.
References:
- N. P. Herzberg;
On a problem of Hurwitz.
Pacific J. Math. 50 (1974) 485-493.
MR 50#233
- Richard K. Guy;
Unsolved Problems In Number Theory, second edition, Springer Verlag,
Berlin, Second edition, 1994.
D 12: Markoff numbers
- Karl Scherer;
Sums of Squares and Products.
J. of Recreational Mathematics 26:4 302 Question 2190 by Karl Scherer
J. of Recreational Mathematics 27:4 (1995) 315-318 partial solution to Q 2190