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 4 of 5: Multiply complete squares

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In plain words

Multiply complete squares

N_3=N_1N_2,\qquad 1\le k\le K_1K_2

Detailed analysis

Call $N_i$ complete when every $1\le k\le K_i=N_i-14$ is representable. If $N_1$ uses i squares and each of those i blocks is replaced by a representation of $N_2$ using a chosen number of squares, the product uses every count up to $1\le k\le K_1K_2$ because the intervals from i to iK_2 overlap.