Problem 11894
(American Mathematical Monthly, Vol.123, March 2016) Proposed by E. J. Ionascu (USA).
Let a, b, c, and d be integers such that a2+ b2+ c2= d2and d 6= 0. Let x, y, and z be three integers such that ax + by + cz = 0.
(a) Prove that x2+ y2+ z2can be written as the sum of two squares.
(b) Let ABCD be a square in R3 with integer vertices A, B, C, and D. Show that the side lengths of ABCD have the form√
l, where l is the sum of two squares.
Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.
(a) Without loss of generality we may assume that gcd(a, b, c) = 1, c is odd and d > 0. Then by [1, Theorem 1] there are integers t1, t2, t3, t4 such that
a= 2(t1t3− t2t4), b= 2(t1t4+ t2t3), c= t21+ t22− t23− t24, d= t21+ t22+ t23+ t24. Let
U= (t21−t22−t23+t24,2(t1t2−t3t4), −2(t1t3+t2t4)), V = (−2(t1t2+t3t4), t21−t22+t23−t24,−2(t1t4−t2t3)), then it can be easily verified that
|U| = |V| = d, U⊥ V, and U× V = d(a, b, c).
Therefore the vectors (x, y, z), U, V are in the plane passing through the origin of normal (a, b, c).
So we can express the vector (x, y, z) as a linear combination of the orthogonal basis given by U, V:
(x, y, z) = c1
d2U+ c2
d2V
with c1= (x, y, z) • U ∈ Z, c2= (x, y, z) • V ∈ Z, which implies that d2(x2+ y2+ z2) = c21+ c22. Finally we use the fact that a positive integer is a sum of two squares if and only if every prime of the form 4k + 3 in its prime power factorization occurs to an even power.
Now d2(x2+ y2+ z2) is written as the sum of two squares, and therefore any prime of the form 4k + 3 in its prime power factorization occurs to an even power. It follows that the same is true for x2+ y2+ z2, which means that also x2+ y2+ z2 can be written as the sum of two squares.
(b) Let u = (u1, u2, u3) = B − A, v = (v1, v2, v3) = D − A and w = (w1, w2, w3) = u × v . Then ui, vi, wi ∈ Z for i = 1, 2, 3 and l = |u|2= |v|2∈ Z. Therefore,
w12+ w22+ w32= |u|2|v|2= l2 and u1w1+ u2w2+ u3w3= 0,
which, by (a), imply that l = u21+ u22+ u23 can be written as the sum of two squares. References
[1] R. Spira, The Diophantine Equation x2+ y2+ z2 = m2, Amer. Math. Monthly 69 (1962), 360-365.