\[p_1, \; p_2, \; p_3\]
with each prime greater than 3, form an arithmetic progression. The common difference must be a multiple of 2, since each prime is odd. If the common difference \[d\]
is not a multiple of 3, so \[d=3k+1\]
or \[3k+2\]
. \[p_1\]
is not a multiple of 3 either, so \[p_1=3j+1\]
or \[3j+2\]
.\[d, \; p_1 \ p_2, \; p_3\] |
\[p_2=p_1+d\] |
\[p_3=p_1+2d\] |
|
\[d=3k+1, \; p_1=3j+1\] |
\[3k+3j+1+2\] |
\[6k+3j+3\] |
|
\[d=3k+2, \; p_1=3j+1\] |
\[3k+3j+3\] |
\[6k+3j+5\] |
|
\[d=3k+1, \; p_1=3j+2\] |
\[3k+3j+3\] |
\[6k+3j+4\] |
|
\[d=3k+2, \; p_1=3j+2\] |
\[3k+3j+4\] |
\[6k+3j+6\] |
\[p_2\]
or \[p_3\]
is divisible by 3, contradicting that they are prime, so \[d\]
is a multiple of 2 and 3 i.e. 6.In general for a sequence of
\[m\]
primes, the common difference must be a multiple of \[m!\]
.