Verify the following:
(a) There exist infinitely many primes ending in 33, such as 233, 433, 733, 1033,….
[Hint: Apply Dirichlet’s theorem.]
(b) There exist infinitely many primes that do not belong to any pair of twin primes.
[Hint: Consider the arithmetic progression 21k + 5 for k= 1,2,….]
(c) There exists a prime ending in as many consecutive 1’s as desired.
[Hint: To obtain a prime ending in n consecutive 1’s, consider the arithmetic progression 10nk + Rn for k= 1,2,….]
(d) There exist infinitely many primes that contain but do not end in the block of digits 123456789.
[Hint: Consider the arithmetic progression 1011k + 1234567891 for k = 1,2,….]
We need at least 10 more requests to produce the solution.
0 / 10 have requested this problem solution
The more requests, the faster the answer.