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# Number Theory of Primes | AIME I, 2015

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 based on Number Theory of Primes.

## Number Theory of Primes - AIME 2015

There is a prime number p such that 16p+1 is the cube of a positive integer. Find p.

• is 307
• is 250
• is 840
• cannot be determined from the given information

### Key Concepts

Series

Theory of Equations

Number Theory

## Check the Answer

AIME, 2015

Elementary Number Theory by Sierpinsky

## Try with Hints

First hint

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

Second Hint

p not equal to a then we have a=16,

Final Step

p\(=16^{2}+3(16)+3=307

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