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 ?
------------------------------------------------------
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