On a problem of Terence Tao

TL;DR

This paper addresses a number theory problem posed by Terence Tao, establishing a lower bound on primes within a specific interval that satisfy certain composite conditions involving linear and exponential expressions.

Contribution

It proves a new lower bound on the count of primes meeting complex algebraic conditions, advancing understanding of prime distribution in structured sets.

Findings

Establishes a lower bound proportional to N/ log N for primes with specified properties.

Demonstrates the existence of many primes satisfying complex algebraic constraints.

Provides a method to count primes in intervals with prescribed algebraic conditions.

Abstract

In this paper, we solve a problem of Terence Tao. We prove that for any $K\geq 2$ and sufficiently large $N$, the number of primes $p$ between $N$ and $(1+\frac{1}{K})N$ such that $\mid kp+ja^{i}+l\mid$ is composite for all $1\leq a, |j|, k\leq K$, $1\leq i \leq K\log N$ and $l$ in any set $L =L_{N}\subseteq\{-KN,\cdots, KN\}$ of cardinality $K$ with $ja^{i}+l\neq0$ is at least $C_{K}\frac{N}{\log N}$, where $C_{K}>0$ depending only on $K$.