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

Exceptional characters and prime numbers in sparse sets