(a) Prove that x2+ y2+ z2can be written as the sum of two squares

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.