Lemoine's conjecture

Lemoine's conjecture states that every odd integer greater than 5 is the sum of an odd prime and twice a prime.

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Goldbach's conjecture: Goldbach's conjecture states that every even integer greater than 2 is the sum of two primes. It is a closely related additive prime conjecture with a different parity pattern.
Émile Lemoine: Émile Lemoine was a French mathematician known for work in geometry and number theory. His name became attached to the conjecture about odd integers and prime summands.
Odd number: An odd integer is an integer not divisible by 2. The conjecture concerns odd targets and specifically requires an odd prime summand.
Chen's theorem: Chen's theorem states that every sufficiently large even integer is a prime plus a number with at most two prime factors. It proves a related additive result with a weaker condition on one summand.
Prime-counting function: The prime-counting function π(x) gives the number of primes no greater than x. Counting candidate primes helps estimate how many decompositions an integer may have.
Additive number theory: Additive number theory studies representations of integers as sums of elements from specified sets. Lemoine's conjecture is a prime-representation problem within this field.
Even number: An even integer is an integer divisible by 2. Twice a prime is always even, so subtracting it leaves an odd prime candidate.
Weak Goldbach conjecture: The weak Goldbach conjecture states that every odd integer greater than 5 is the sum of three primes. It has the same odd targets but uses three primes instead of a prime plus twice a prime.
Hardy–Littlewood prime tuples conjecture: A conjectural framework estimating the frequency of prime patterns defined by linear forms. Its methods predict frequencies for additive representations involving primes.
Prime number: A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. The conjecture's two summands are required to be prime.