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 5 of 5: Fill the remaining high counts

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In plain words

Fill the remaining high counts

K_1K_2>\frac23N_1N_2

Detailed analysis

For $N\ge169$ , every excess $d\ge14$ can be written as $d=3a+8b$ ; starting from ones and grouping four or nine equal squares therefore covers all counts above $N/2$ through $N-14$ . Since $K_i=N_i-14>\frac23N_i$ , the product range overlaps this high-count range. Repeating products from $13^2$ gives infinitely many $13^{2r}$ with $S(13^r)=13^{2r}-14$ .