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Number Theory Level 2

It is 5 100 + 3 100 5^{100}+3^{100} a perfect square?

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Anca Baltariga by Anca Baltariga , 26, Romania

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2 solutions

Jaydee Lucero
Jul 6, 2017

Assume that n 2 = 5 100 + 3 100 n^2 = 5^{100} + 3^{100} , where n n is a positive integer. Then it must satisfy n 2 ≡ 0 or 1 ( m o d 4 ) . n^2 \equiv \color{#D61F06}{0}\text{ or }\color{#D61F06}{1} \color{#333333}{\pmod{4}}. But 5 100 + 3 100 ≡ 1 100 + ( − 1 ) 100 = 1 + 1 = 2 ( m o d 4 ) , 5^{100} + 3^{100} \equiv 1^{100} + (-1)^{100} = 1 + 1 = \color{#D61F06}{2} \color{#333333}{\pmod{4}}, a contradiction. Thus, 5 100 + 3 100 5^{100} + 3^{100} is not a perfect square .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Hint : Consider modulo 8.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

I took modulo 4.

Spandan Senapati - 3 years, 11 months ago

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I am also wondering why I took mod8 :P

Harsh Shrivastava - 3 years, 11 months ago

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Nvr mind.That's the beauty of Maths

Spandan Senapati - 3 years, 11 months ago

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