Problem 6

For each positive integer

n

, let

S(n)

be the greatest integer such that every

k\le S(n)

permits writing

n^2

as a sum of

k

positive squares. (a) Prove

S(n)\le n^2-14

for

n\ge4

; (b) find

n

with equality; (c) prove infinitely many such

n

Step 1 of 5: Prove the universal upper bound

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In plain words

Prove the universal upper bound

13=3a+8b+15c+\cdots

Detailed analysis

Put $N=n^2$ . A representation of N with $N-13$ positive squares would have excess $13$ over that many ones. The available excesses are $3,8,15,\ldots$ , so $13=3a+8b+15c+\cdots$ . Since $15>13$ , this reduces to $13=3a+8b$ , which has no solution. Thus $N-13$ is impossible and $S(n)\le N-14$ .