We prove that almost surely the first non-empty (k+1)-core to arise during the random graph process will have a k-factor or will be k-factor-critical. Thus the threshold for the appearance of a k-regular subgraph is at most the threshold for the (k+1)-core. This improves a result of Pralat, Verstraete and Wormald [5] and proves a conjecture of Bollobas, Kim and Verstraete [3].

Let $$n\ne 0$$ be an integer. A set of m distinct positive integers $$\{a_1,a_2,\ldots ,a_m\}$$ is called a D(n)-m-tuple if $$a_ia_j + n$$ perfect square for all $$1\le i < j \le m$$ . k In this paper, we prove that $$\{k,k+1,c,d\}$$ $$D(-k)$$ -quadruple with $$c>1$$ , then $$d=1$$ The proof relies not only on standard methods in field (Baker’s linear forms logarithms and the hypergeometric met...