Émile Lemoine notable work Lemoine's conjecture
Lemoine's conjecture and extensions
In 1894, Lemoine stated what is now known as Lemoine's conjecture: Every odd number which is greater than three can be expressed in the form 2p + q where p and q are prime. In 1985, John Kiltinen and Peter Young conjectured an extension of the conjecture which they called the "refined Lemoine conjecture". They published the conjecture in a journal of the Mathematical Association of America:  "For any odd number m which is at least 9, there are odd prime numbers p, q, r and s and positive integers j and k such that m = 2p + q, 2 + pq = 2j + r and 2q + p = 2k + s. [...] the study has directed our attention to more subtle aspects of the additive theory of prime numbers. Our conjecture reflects this, dealing with interactions of sums involving primes whereas Goldbach's conjecture and Lemoine's conjecture deal with such sums only individually. This conjecture and the open questions about numbers at levels two and three are of interest in their own right because of the issues they raise within this fascinating and often baffling additive realm of the prime numbers."
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