Thread: Goldbach Conjecture

Any even number greater than or equal to 4 can be written as the sum of two primes.

Is this conjecture true if we replace "even" by "odd"?

Is this conjecture true if we replace "sum" by "product"?

Originally Posted by MATNTRNG

Any even number greater than or equal to 4 can be written as the sum of two primes.

Is this conjecture true if we replace "even" by "odd"?

Is this conjecture true if we replace "sum" by "product"?

Since 2 is the only even prime, we get the first counterexample when we have two composite odd numbers in a row... namely, 27.

For the second question just take 2*3*5 as counterexample... or 2*2*2 for that matter.

EDIT: Actually the first counterexample for the "odd" problem is 11, I wasn't thinking clearly when I wrote 27. (27 is the smallest odd number greater than 4 that cannot be written as the sum of at most 2 primes.)