Running through the positive integers in turn: {{begin-eqn}} {{eqn | l = 6 \times 1 - 1       | r = 5       | c = which is prime }} {{eqn | l = 6 \times 1 + 1       | r = 7       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 2 - 1       | r = 11       | c = which is prime }} {{eqn | l = 6 \times 2 + 1       | r = 13       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 3 - 1       | r = 17       | c = which is prime }} {{eqn | l = 6 \times 3 + 1       | r = 19       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 4 - 1       | r = 23       | c = which is prime }} {{eqn | l = 6 \times 4 + 1       | r = 25 = 5^2       | c = and so is composite }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 5 - 1       | r = 29       | c = which is prime }} {{eqn | l = 6 \times 5 + 1       | r = 31       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 6 - 1       | r = 35 = 5 \times 7       | c = and so is composite }} {{eqn | l = 6 \times 5 + 1       | r = 37       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 7 - 1       | r = 41       | c = which is prime }} {{eqn | l = 6 \times 7 + 1       | r = 43       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 8 - 1       | r = 47       | c = which is prime }} {{eqn | l = 6 \times 8 + 1       | r = 49 = 7^2       | c = and so is composite }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 9 - 1       | r = 53       | c = which is prime }} {{eqn | l = 6 \times 9 + 1       | r = 55 = 5 \times 11       | c = and so is composite }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 10 - 1       | r = 59       | c = which is prime }} {{eqn | l = 6 \times 10 + 1       | r = 61       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 11 - 1       | r = 65 = 5 \times 13       | c = and so is composite }} {{eqn | l = 6 \times 11 + 1       | r = 67       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 12 - 1       | r = 71       | c = which is prime }} {{eqn | l = 6 \times 12 + 1       | r = 73       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 13 - 1       | r = 77 = 7 \times 11       | c = and so is composite }} {{eqn | l = 6 \times 13 + 1       | r = 79       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 14 - 1       | r = 83       | c = which is prime }} {{eqn | l = 6 \times 14 + 1       | r = 85 = 5 \times 17       | c = and so is composite }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 15 - 1       | r = 89       | c = which is prime }} {{eqn | l = 6 \times 15 + 1       | r = 91 = 7 \times 13       | c = and so is composite }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 16 - 1       | r = 95 = 5 \times 19       | c = and so is composite }} {{eqn | l = 6 \times 16 + 1       | r = 97       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 17 - 1       | r = 101       | c = which is prime }} {{eqn | l = 6 \times 17 + 1       | r = 103       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 18 - 1       | r = 107       | c = which is prime }} {{eqn | l = 6 \times 18 + 1       | r = 109       | c = which is prime }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 19 - 1       | r = 113       | c = which is prime }} {{eqn | l = 6 \times 19 + 1       | r = 115 = 5 \times 23       | c = and so is composite }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 6 \times 20 - 1       | r = 119 = 7 \times 17       | c = and so is composite }} {{eqn | l = 6 \times 20 + 1       | r = 121 = 11^2       | c = and so is composite }} {{end-eqn}} {{qed}}