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 3 of 5: Cover the other two residue classes

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In plain words

Cover the other two residue classes

169=5\cdot4^2+149\cdot1^2,\quad169=9^2+9^2+151\cdot1^2

Detailed analysis

The representations $169=5\cdot4^2+149\cdot1^2$ and $169=9^2+9^2+151\cdot1^2$ cover the two remaining residue classes by the same grouping operation. The explicit decompositions listed for counts 7,4,3,6 fill the short tails, so every $1\le k\le155$ occurs and $S(13)=155=13^2-14$ .