# Goldbach's conjecture

Goldbach's conjecture: Every even number > 4 is the sum of two odd primes.

• 6=3+3, 8=3+5, 10=3+7=5+5, 12=5+7, 14=3+11=7+7, 16=3+13=5+11, 18=5+13=7+11, 20=3+17=7+13, 22=3+19=5+17=11+11, 24=5+19=7+17=11+13, 26=3+23=7+19=13+13, 28=5+23=11+17, 30=7+23=11+19=13+17, 32=3+29=13+19, 34=3+31=5+29=11+23=17+17, 36=5+31=7+29=13+23=17+19, 38=7+31=19+19, 40=3+37=11+29=17+23, 42=5+37=11+31=13+29=19+23, 44=3+41=7+37=13+31, 46=3+43=5+41=17+29=23+23, 48=5+43=7+41=11+37=17+31=19+29, 50=3+47=7+43=13+37=19+31, 52=5+47=11+41=23+29, 54=7+47=11+43=13+41=17+37=23+31, ...
• 389,965,026,819,938 = 5,569 + 389,965,026,814,369 (and no decomposition with a smaller prime exists)
• Unproved, but believed to be true. Tested true up to 1017
• $g(n) =\# \{\, p,q : \mbox{prime} \mid n=p+q \land p \leq q \,\}$ is the number of distinct ways an even number can be so partitioned

Goldbach's odd (or weak) conjecture: Every odd number > 7 is the sum of three odd primes.

• decompositions easily generated from the even decompositions, by systematically subtracting primes
• 9=3+3+3, 11=3+3+5, 13=3+3+7=3+5+5, 15=3+5+7=5+5+5, 17=3+3+11=3+7+7=5+5+7, ...
• Proved under the assumption of the truth of the generalized Riemann hypothesis; proved for all "sufficiently large" numbers