Sophie Germain?

Number Theory Level 4

Find the sum of all positive integral a < 2014 a<2014 such that a x 4 + 1 ax^4+1 can be factored to ( b x 2 − c x + 1 ) ( b x 2 + c x + 1 ) (bx^2-cx+1)(bx^2+cx+1) for all x x where b , c b,c are integers.


The answer is 1416.

Trevor Arashiro by Trevor Arashiro , 22, USA

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1 solution

Pranjal Jain
Dec 11, 2014

( b x 2 + c x + 1 ) ( b x 2 − c x + 1 ) (bx^2+cx+1)(bx^2-cx+1) = b 2 x 4 + ( 2 b − c 2 ) x 2 + 1 =b^2x^4+(2b-c^2)x^2+1

By comparing coefficient of x 4 x^4 , a = b 2 a=b^2

Now, comparing coefficient of x 2 x^2 , 2 b = c 2 2b=c^2 . So b b must be of the form 2 k 2 2k^2 . Hence, a a would be of the form ( 2 k 2 ) 2 = 4 k 4 (2k^2)^2=4k^4 .

  • k = 1 , a = 4 k=1,\ a=4
  • k = 2 , a = 64 k=2,\ a=64
  • k = 3 , a = 324 k=3,\ a=324
  • k = 4 , a = 1024 k=4,\ a=1024
  • k = 5 , a = 2500 k=5,\ a=2500 (Wrong case, exceeds 2014) \color{#D61F06}{\text{ (Wrong case, exceeds 2014)}}

Therefore, required result is 4 + 64 + 324 + 1024 = 1416 4+64+324+1024=\boxed{1416}

9 Helpful 0 Interesting 0 Brilliant 0 Confused

Yup, exactly how I did it, although I think it would be better if u used like y or something because a and α \alpha look too similar.

Trevor Arashiro - 6 years, 6 months ago

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Thanks for your feedback! Edited!

Pranjal Jain - 6 years, 6 months ago

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Yes, that looks much better. Thanks!

Trevor Arashiro - 6 years, 6 months ago

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