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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 termis given by $a_k=(k-1)(q-p)+p$ (i.e., with $a_1=p$ and $a_2=q$). Here weconjecture 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 ofthe first $2n$ terms of the arithmetic progression $A(p,q)$. Notice that in thecase of even $a$, this conjecture immediately follows from Goldbach'sconjecture. We also propose the analogous conjecture for odd positive integers$a>1$ as well as some related Goldbach's like conjectures arising from thepreviously mentioned arithmetic progressions.

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