Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 based on Number Theory of Primes.
There is a prime number p such that 16p+1 is the cube of a positive integer. Find p.
Series
Theory of Equations
Number Theory
Answer: is 307.
AIME, 2015
Elementary Number Theory by Sierpinsky
Notice that 16p+1must be in the form \((a+1)^{3}=a^{3}+3a^{2}+3a\), or \(16p=a(a^{2}+3a+3)\). Since p must be prime, we either have p=a or a=16
p not equal to a then we have a=16,
p\(=16^{2}+3(16)+3=307
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