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Number Theory of Primes | AIME I, 2015

Try this beautiful problem from American Invitational Mathematics Examination, AIME, 2009 based on geometric sequence. Use hints to solve the problem.

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


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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