Editing Goldbach's conjecture (section)

== Formal statement ==
Each of the three conjectures has a natural analog in terms of the modern definition of a prime, under which 1 is excluded. A modern version of the first conjecture is:
{{block indent|Every integer that can be written as the sum of two primes can also be written as the sum of as many primes as one wishes, until either all terms are two (if the integer is even) or one term is three and all other terms are two (if the integer is odd).}}
A modern version of the marginal conjecture is:

{{block indent|Every integer greater than 5 can be written as the sum of three primes.}}

And a modern version of Goldbach's older conjecture of which Euler reminded him is:

{{block indent| Every even integer greater than 2 can be written as the sum of two primes.}}

These modern versions might not be entirely equivalent to the corresponding original statements. For example, if there were an even integer {{math|''N'' {{=}} ''p'' + 1}} larger than 4, for {{mvar|p}} a prime, that could not be expressed as the sum of two primes in the modern sense, then it would be a counterexample to the modern version of the third conjecture (without being a counterexample to the original version). The modern version is thus probably stronger (but in order to confirm that, one would have to prove that the first version, freely applied to any positive even integer {{mvar|n}}, could not possibly rule out the existence of such a specific counterexample {{mvar|N}}). In any case, the modern statements have the same relationships with each other as the older statements did. That is, the second and third modern statements are equivalent, and either implies the first modern statement.

The third modern statement (equivalent to the second) is the form in which the conjecture is usually expressed today. It is also known as the "[[Mathematical jargon#strong|strong]]", "even", or "binary" Goldbach conjecture. A weaker form of the second modern statement, known as "[[Goldbach's weak conjecture]]", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture", asserts that 

{{block indent|Every odd integer greater than 7 can be written as the sum of three odd primes.}}