# On a Conjecture of Cai-Zhang-Shen for Figurate Primes

## TL;DR

This paper proves the Cai-Zhang-Shen conjecture that every integer greater than one can be expressed as the sum of two figurate primes, using advanced mathematical techniques including Lagrange multipliers and Cardano's formula.

## Contribution

The paper provides a rigorous proof of the conjecture, introducing an equivalent formulation and employing optimization and algebraic methods.

## Key findings

- The conjecture is proven true for all integers greater than one.
- The proof involves solving constrained optimization problems with Lagrange multipliers.
- Algebraic methods like Cardano's formula are used to solve cubic equations in the proof.

## Abstract

A conjecture of Cai-Zhang-Shen for figurate primes says that every integer $k>1$ is the sum of two figurate primes. In this paper we give an equivalent proposition to the conjecture. By considering extreme value problems with constraints about the conjecture in the cases of odd and even integers and using the method of Lagrange multipliers, Cardano formula for cubic equations and the contradiction, we prove the conjecture.