# Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes

## TL;DR

This paper introduces conjectures extending Goldbach's conjecture by exploring sums of terms from specific arithmetic progressions starting with primes, suggesting these sums can represent all integers greater than one.

## Contribution

It proposes new Goldbach-like conjectures involving sums of arithmetic progression terms defined by prime pairs, expanding the scope of prime-based additive conjectures.

## Key findings

- Conjecture that every integer greater than one can be expressed as a sum of an even number of terms from certain prime-based arithmetic progressions.
- Connection established between these conjectures and the classical Goldbach's conjecture for even integers.
- Introduction of analogous conjectures for odd integers and related prime sum problems.

## Abstract

For two odd primes $p$ and $q$ such that $p<q$, let $A(p,q):=(a_k)_{k=1}^{\infty}$ be the arithmetic progression whose $k$th term is given by $a_k=(k-1)(q-p)+p$ (i.e., with $a_1=p$ and $a_2=q$). Here we conjecture that for every positive integer $a>1$ there exist a positive integer $n$ and two odd primes $p$ and $q$ such that $a$ can be expressed as a sum of the first $2n$ terms of the arithmetic progression $A(p,q)$. Notice that in the case of even $a$, this conjecture immediately follows from Goldbach's conjecture. We also propose the analogous conjecture for odd positive integers $a>1$ as well as some related Goldbach's like conjectures arising from the previously mentioned arithmetic progressions.