Difference between revisions of "Sophie Germain Identity"
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== Problems == | == Problems == | ||
=== Introductory === | === Introductory === | ||
| + | *What is the remainder when <math>2^{202} +202</math> is divided by <math>2^{101}+2^{51}+1</math>? (2020 AMC10 B) | ||
*Prove that if <math>n>1</math> then <math>n^4 + 4^n</math> is [[composite]]. (1978 Kurschak Competition) | *Prove that if <math>n>1</math> then <math>n^4 + 4^n</math> is [[composite]]. (1978 Kurschak Competition) | ||
Revision as of 19:04, 28 October 2023
The Sophie Germain Identity states that:
One can prove this identity simply by multiplying out the right side and verifying that it equals the left. To derive the factoring, we begin by completing the square and then factor as a difference of squares:
Problems
Introductory
- What is the remainder when
is divided by 
? (2020 AMC10 B)
is divided by 
? (2020 AMC10 B)- Prove that if
then 
is composite. (1978 Kurschak Competition)
then 
is Intermediate
- Compute
. (1987 AIME, #14)
. (- Find the largest prime divisor of
. (Mock AIME 5 2005-2006 Problems/Pro)
. (Mock AIME 5 2005-2006 Problems/Pro)- Calculate the value of
. (BMO 2013 #1)
. (BMO 2013 #1)- Find the largest prime factor of
, given that it is the product of three distinct primes. (ARML 2016 Individual #10)
, given that it is the product of three distinct primes. (ARML 2016 Individual #10)
