A number theory problem by Suryadeep Bandyopadhyay

Number Theory Level 3

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


The answer is 307.

Suryadeep Bandyopadhyay by Suryadeep Bandyopadhyay , 22, India

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1 solution

Eziz Hudaykulyyev
Jan 19, 2016

16p+1=x^3 we can write as 16p=x^3-1^3

16p=(x-1)(x^2+x+1)

16=x-1 → x=17

p=x^2+x+1 → p=307

1 Helpful 0 Interesting 0 Brilliant 0 Confused

How did you get that 16=x-1 after splitting x^3-1^3

Joy Patel - 5 years, 4 months ago

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Because x^2+x+1 is odd

Tran Hieu - 5 years, 4 months ago

If 16 is equal to x^2+x+1, thus x will not be whole number.

Eziz Hudaykulyyev - 5 years, 4 months ago

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