MathDB

1

Part of 2005 Poland - Second Round

Problems(2)

Integer m exists such that W(m)=W(m+1)=0

Source:

12/6/2010
The polynomial W(x)=x2+ax+bW(x)=x^2+ax+b with integer coefficients has the following property: for every prime number pp there is an integer kk such that both W(k)W(k) and W(k+1)W(k+1) are divisible by pp. Show that there is an integer mm such that W(m)=W(m+1)=0W(m)=W(m+1)=0.
algebrapolynomialmodular arithmeticquadraticsnumber theory proposednumber theory
Both expressions are primes

Source:

12/6/2010
Find all positive integers nn for which nn+1n^n+1 and (2n)2n+1(2n)^{2n}+1 are prime numbers.
number theory proposednumber theory