Problem
NT-B2-M09-P019 Many Composite Values of Linear Expressions
Let \(a_1,\ldots,a_k\) be distinct integers. Prove that there are infinitely many \(N\) such that all numbers \(N+a_1,\ldots,N+a_k\) are composite.
Choose distinct primes \(p_i\) not dividing the differences \(a_i-a_j\), and impose \(N+a_i\equiv 0\pmod {p_i}\).
Choose pairwise distinct primes \(p_1,\ldots,p_k\) so large that \(p_i>|a_i-a_j|\) for all \(i,j\). The system \(N\equiv -a_i\pmod {p_i}\) is solvable by CRT. Let \(N_0\) be a solution and \(P=p_1\cdots p_k\). Then \(N=N_0+tP\) preserves all divisibilities \(p_i\mid N+a_i\). For sufficiently large \(t\), all \(N+a_i\) are positive and larger than \(p_i\), hence composite. There are infinitely many such \(t\).
A nonstandard CRT application; the choice of large primes should be made explicit.