Goldbach’s conjecture in max-algebra

Abstract

The Goldbach conjecture is one of the best known open problems in number theory. It claims that every even integer greater than 2 can be written as the sum of two primes. The present paper formulates a max-algebraic claim that is equivalent to Goldbach’s conjecture. The max-algebraic analogue allows examination of the conjecture by the methods of max-algebra. A max-algebra is an algebraic structure in which classical addition $$+$$ and multiplication $$\times $$ are replaced by the operations maximum $$\oplus $$ and addition $$\otimes $$ , in other words $$a\oplus b=\max \{a,b\}$$ and $$a\otimes b=a+b$$ .