primes in arithmetic progressions

[David Wasserman]

I have made a conjecture that every prime p is the start of p primes  
in arithmetic progression.  For example,
2, 3
3, 5, 7
5, 11, 17, 23, 29
7, 157, 307, 457, 607, 757, 907
11, 1536160091, 3072320171, 4608480251, 6144640331, 7680800411,  
9216960491, 10753120571, 12289280651, 13825440731, 15361600811
Does anyone have any ideas about proving or disproving this?  Does it  
follow from a well-known conjecture?