MathDB

2

Part of 1990 Irish Math Olympiad

Problems(2)

Greatest Prime divisor problem

Source: Irish MO

5/10/2012
A sequence of primes ana_n is defined as follows: a1=2a_1 = 2, and, for all n \geq 2,an a_n is the largest prime divisor of a1a2...an−1+1a_1a_2...a_{n-1} + 1. Prove that an≠5a_n \neq 5 for all n. I'm presuming it must involve proving it's never equal to 0 mod 5, but I don't know what to do. Thanks
number theory unsolvednumber theory
15 Prime Numbers in Arithmetic Progression

Source: 1990 IrMO Paper 2 Problem 2

9/30/2017
Suppose that p1<p2<⋯<p15p_1<p_2<\dots <p_{15} are prime numbers in arithmetic progression, with common difference dd. Prove that dd is divisible by 2,3,5,7,112,3,5,7,11 and 1313.
number theoryprime numbersarithmetic sequence