primes of the form 10^k*5080978447+1
 

k=1
is prime
k=29 is prime

any other prime of the form 10^k*5080978447+1?

1
9
10
19
29
49
476
2009

As an aside I would write this as 5080978447*10^k+1. When I first read it I thought you were multiplying the exponent by the big number!

[QUOTE=lukerichards;515430]As an aside I would write this as 5080978447*10^k+1. When I first read it I thought you were multiplying the exponent by the big number![/QUOTE]It took me a moment to decipher the rendition in the OP also.

It did give me an excuse to see how sieving speeds things up in an actual example. I first ran a totally mindless script verifying the exponents giving (pseudo)primes up to the limit 3000. (Previously-stated results confirmed.)

Then, I determined the possible prime factors p of n = 5080978447*10^k + 1 and looked at the number of n's not divisible by any of these p, up to a modest limit. The number of candidates dropped as the limit on p was raised. Each candidate excluded by having a small factor, is a humungous number not being subjected to a pseudoprime test. Sieving out candidates divisible by small primes sped things up considerably!

I leave it as an exercise to give a condition on p, 5 < p < 5080978447, that determines whether p divides 5080978447*10^k + 1 for some positive integer k. Note that 2, 3, and 5 are automatically excluded as factors of 5080978447*10^k + 1.

If I did my sums right, the values of p < 200 are [7, 13, 17, 19, 23, 29, 31, 43, 47, 59, 61, 71, 79, 83, 89, 97, 107, 109, 113, 131, 149, 151, 163, 167, 179, 181, 191, 193].