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 2 of 5: Reach the bound for

n=13

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In plain words

Reach the bound for $n=13$

169=9+4+4+152\cdot1^2

Detailed analysis

For $N=169$ , the displayed representation has $155=N-14$ terms. Replacing four $1^2$ by one $2^2$ decreases the count by 3; continuing with four $2^2$ and then four $4^2$ gives all counts $38,35,\ldots,11,8,5$ . Finally $169=12^2+5^2$ gives count 2.