Special integers

Number Theory Level 2

Let N N denote a positive integer such that one of { N − 1 , N + 1 } \{N-1, N+1\} is a perfect square while the other is a perfect cube.

What is the sum of all possible values of N N ?


The answer is 26.

Freddie Hand by Freddie Hand , 18, United Kingdom

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2 solutions

Mahdi Raza
May 24, 2020

By observation, When N = 26 ⟹ { N − 1 = 5 2 } , { N + 1 = 3 3 } N= 26 \implies \{N-1 = 5^2\}, \{N+1 = 3^3\}

This is not a rigorous method, if anyone has it's proof please share it!

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How do you put curly brackets in LaTeX?

Vinayak Srivastava - 1 year ago

26 26 is the only number (which I know of) which gives{ 25 , 27 25,27 } where 25 = 5 2 , 27 = 3 3 25=5^2, 27= 3^3 . I am waiting for someone to prove it.

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Even I guessed it

Mahdi Raza - 1 year ago

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