Let$u$_{$n$}be the$n$th term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest prime factor of$u$_{$n$}which is of the form$n$exp(log$n$/104 log log$n$). In doing so, we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdős from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.

We improve upon the local bound in the depth aspect for sup-norms of newforms on D^×, where D is an indefinite quaternion division algebra over Q. Our sup-norm bound implies a depth-aspect subconvexity bound for L(1/2,f×θ_χ), where f is a (varying) newform on D^× of level p^n, and θ_χ is an (essentially fixed) automorphic form on GL_2 obtained as the theta lift of a Hecke character χ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via p-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

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In this paper we prove the Iwasawa main conjecture for the Rankin-Selberg product of a general modular form and a CM form of higher weight. This result has been used by others to prove the converse of the theorem of Gross-Zagier and Kolyvagin, and the BSD formula for elliptic curves of analytic rank one.

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