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March 22, 2020

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


Answer: is 307.

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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