MathDB

10

Part of PEN P Problems

Problems(1)

P 10

Source:

5/25/2007
For each positive integer  n,  S(n) \,n,\;S(n)\, is defined to be the greatest integer such that, for every positive integer  k≤S(n),  n2 \,k\leq S(n),\;n^{2}\, can be written as the sum of  k \,k\, positive squares. [*] Prove that S(n)≤n2−14S(n)\leq n^{2}-14 for each n≥4n\geq 4. [*] Find an integer nn such that S(n)=n2−14S(n)=n^{2}-14. [*] Prove that there are infinitely many integers nn such that S(n)=n2−14S(n)=n^{2}-14.
Additive Number Theory