Sophie Germain Identity: Factoring a^4+4b^4
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- Sophie Germain Identity One of the cleverest, most useful identities I've seen is the Sophie Germain Identity. Problem: Factor a4+4b4a^4+4b^4a4+4b4. Solution: Factoring these si…
By Cody Johnson, Joshua Siktar
- Sophie Germain Identity is a very common and useful tactic in number theory. It is basically just a factorization identity. \begin{eqnarray*} a^4+4b^4 & = & (a^2+2ab+2b^2)(a^2-2…
By amaterasu, Joshua Siktar
- We are going to derive the Sophie Germain Identity. It seems quite easy to multiply the factorization after you have been given the factorization, but how does one find it in th…
By Wayne Zhao, Joshua Siktar
Problems 4
- Find all positive integers aaa and bbb such that a4+4b4a^4+4b^4a4+4b4 is prime. Hint: The Sophie Germain Identity allows you to factorize a4+4b4a^4+4b^4a4+4b4.
- List all prime numbers ppp for which p−4p-4p−4 is the 444th power of an integer.
- (1987 AIME #14) Compute: (104+324)(224+324)(344+324)(464+324)(584+324)(44+324)(164+324)(284+324)(404+324)(524+324).\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4…
- The number 104,060,465104{,}060{,}465104,060,465 is divisible by a five-digit prime number. What is that prime number? (Hint: Do the digits look like a row of Pascal's triangle?…