  We develop a lower bound sieve for primes under the (unlikely) assumption of
infinitely many exceptional characters. Compared with the illusory sieve due to
Friedlander and Iwaniec which produces asymptotic formulas, we show that less
arithmetic information is required to prove non-trivial lower bounds. As an
application of our method, assuming the existence of infinitely many
exceptional characters we show that there are infinitely many primes of the
form $a^2+b^8$.
