MathDB

4

Part of 2017 Bundeswettbewerb Mathematik

Problems(2)

Recursion And Primes

Source: Bundeswettbewerb Mathematik 2017, Round 1 - #4

8/7/2017
The sequence a0,a1,a2,…a_0,a_1,a_2,\dots is recursively defined by a_0 = 1   \text{and}   a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right)   \text{for } n \geq 1. Prove for each integer n≥1n \geq 1:
(a) The number ana_n is a positive integer. (b) Each prime pp with n<p≤2nn < p \leq 2n is a divisor of ana_n. (c) If nn is a prime, then an−2a_n-2 is divisible by nn.
number theorynumber theory unsolvedSequencerecursionprime numbersInteger
Consecutive Integers As Sum Of Square And Cube

Source: Bundeswettbewerb Mathematik 2017, Round 2 - #4

9/3/2017
We call a positive integer heinersch if it can be written as the sum of a positive square and positive cube. Prove: There are infinitely many heinersch numbers hh, such that h−1h-1 and h+1h+1 are also heinersch.
number theorysquarecube