MathDB

2

Part of 2000 Hungary-Israel Binational

Problems(2)

binomial coeffcient and relation divisible

Source: 11-th Hungary-Israel Binational Mathematical Competition 2000

4/20/2007
Prove or disprove: For any positive integer kk there exists an integer n>1n > 1 such that the binomial coeffcient (ni)\binom{n}{i} is divisible by kk for any 1≤i≤n−1.1 \leq i \leq n-1.
algebrabinomial theoremnumber theory unsolvednumber theory
on $S = \{m^2 + dn^2 \}

Source: 11-th Hungary-Israel Binational Mathematical Competition 2000

4/22/2007
For a given integer dd, let us define S={m2+dn2∣m,n∈Z}S = \{m^{2}+dn^{2}| m, n \in\mathbb{Z}\}. Suppose that p,qp, q are two elements of SS , where pp is prime and p∣qp | q. Prove that r=q/pr = q/p also belongs to SS .
number theory proposednumber theory