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Turkey NMO-2004 P.3, nice problem about diophantine equation

Source:

November 19, 2010

number theory proposednumber theory

Problem Statement

(a) Determine if exist an integer

n

such that

n^2 -k

has exactly

10

positive divisors for each

k = 1, 2, 3.

(b) Show that the number of positive divisors of

n^2 -4

is not

10

for any integer

n.