1600.21 – Even Numbers and Primes

Find a pair of prime numbers that add to 38; to 54; to 104. Is every even number greater than 4 the sum of two primes?

Solution

There are numerous possibilities for each.

38=19+19=7+31,54=11+43=7+47=13+41=17+37=23+31,104=3+101=7+97=31+73=37+67=43+61. \begin{aligned} 38 &=& 19 + 19 = 7 + 31, \\ 54 &=& 11 + 43 = 7 + 47 = 13 + 41 = 17 + 37 = 23 + 31, \\ 104 &=& 3 + 101 = 7 + 97 = 31 + 73 = 37 + 67 = 43 + 61. \end{aligned}

It is conjectured that every\textit{every} even number is the sum of two primes (except 2!). The conjecture is called Goldbach's Conjecture and is one of the most famous unsolved mathematics problem. The conjecture has been shown to be true for all integers less than 4×10184 \times 10^{18} but remains unproven despite considerable effort.