A PROOF OF LEMOINE’S CONJECTURE BY CIRCLES OF PARTITION T. AGAMA AND B. GENSEL Abstract. In this paper we use a new method to study problems in additive number theory. We leverage this method to prove the Lemoine conjecture, a closely related problem to the binary Goldbach conjecture. In particular, we show by using the notion of circles of partition that for all odd numbers n ≥ 9 holds n = p +2q for not necessarily different primes p, q. 1. Introduction Let P and 2P denotes primes numbers and their doubles, respectively. Then Lemoine’s conjecture, roughly speaking, purports all odd numbers can be partition into the set of all prime numbers and their doubles. More formally the conjecture states Conjecture 1.1. The equation 2n +1= p +2q always has a solution in primes p and q (not necessarily distinct) for n> 2. The conjecture was first formulated and posed by Emile Lemoine in 1895 but was wrongly attributed to Hyman Levy in the 1960 (see [1]), which is why it is some- times referred to as Levy’s conjecture. The Lemoine conjecture has not gained much popularity as does the binary Goldbach conjecture but is closely related to and certainly implies the ternary Goldbach conjecture. There has been an amaz- ing computational work in verifying the conjecture, and it is now known that the conjecture holds upto 10 10 [2]. In this paper we apply a method developed in [3] to study the conjecture; In par- ticular, we show that the conjecture holds for all odd numbers n ≥ 9. 2. The Circle of Partition Here we repeat the base results of the method of circles of partition developed in [3]. Definition 2.1. Let n ∈ N and M ⊆ N. We denote with C (n, M)= {[x] | x, y ∈ M,n = x + y} the Circle of Partition generated by n with respect to the subset M. We will abbreviate this in the further text as CoP. We call members of C (n, M) as points and denote them by [x]. For the special case M = N we denote the CoP shortly as Date : March 20, 2021. 2010 Mathematics Subject Classification. Primary 11Pxx, 11Bxx; Secondary 11Axx, 11Gxx. Key words and phrases. Lemoine, circle of partition, axis. 1