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I have a problem to solve, which looks like this: $$a^4+4=b$$

$a$ is a natural number where when it is multiplied by the exponent, it'll be equal to $b$

$b$ is always a prime number

We have to find what number is $a$

First off, every time $a$ is multiplied by the exponent and added by 4, it'll always be an even number no matter what. The problem said that the set for the problem will at least have one number, but I couldn't seem to think of a way it can be so. Any help will be appreciated.

Cheers,

thestrikebacker
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  • Note: I wrote down "root number" on the title of the post, which means something other than I meant. What I actually meant is what number that when has the exponent of 4, will be equal to a prime number when added to four. – thestrikebacker Jun 06 '21 at 11:19
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    Look at this: https://math.stackexchange.com/questions/1121407/show-that-n44-is-not-a-prime-number – Jujustum Jun 06 '21 at 11:19
  • Why (and where) are you multiplying by the exponent? – PM 2Ring Jun 10 '21 at 13:23

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For $a>1$ , $b $ cannot be prime because $a^4 + 4 = (a^2 + 2 + 2a)(a^2+2-2a)$

Note: This is a special case of Sophie German identity which gives factors of $a^4 + 4 c^4$

Infinity_hunter
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