Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 based on Cube of Positive Integer.
There is a prime number p such that 12p+1 is the cube of positive integer.Find p..
Algebra
Theory of Equations
Number Theory
Answer: is 183.
AIME, 2015, Question 3
Elementary Number Theory by David Burton
\(a^{3}=12p+1\) implies that \(a^{3}-1=12p\) that is (a-1)(\(a^{2}\)+a+1)=12p
a is odd, a-1 even, \(a^{2} +a+1 odd implies a-1 multiple of 12 that is here =12 then a=12+1 =13
\(a^{2}+a+1=p implies p= 169+13+1=183.
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