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good catch; added a note about why the condition (a,b) is needed
For example, the sequence of numbers which are one more than a multiple of 11 (1, 12, 23, 34, 45, ...) is an arithmetic progression. Dirichlet's theorem states that this sequence contains infinitely many prime numbers. Equivalently, there are infinitely many values of k for which 11*k+1 is a prime number.
For example, the sequence of numbers which are one more than a multiple of 11 (1, 12, 23, 34, 45, ...) is an arithmetic progression. Dirichlet's theorem states that this sequence contains infinitely many prime numbers. Equivalently, there are infinitely many values of k for which 11*k+1 is a prime number.
The requirement that a and b be relatively prime is crucial: if a and b share a common factor k, we can write a = k*c, b=k*d. Then an+b = kcn+kd=k(cn+d) which can't be prime unless n=0 and d is prime. For example, the arithmetic sequence of integers of the form 12k+2 contains only even numbers, and thus no primes except 2.
Gauss proposed this conjecture but could not prove it. Dirichlet first proved this in 1837 using the Dirichlet L-series. The proof is quite challenging to understand, and is beyond the scope of most number theory courses.
Gauss proposed this conjecture but could not prove it. Dirichlet first proved this in 1837 using the Dirichlet L-series. The proof is quite challenging to understand, and is beyond the scope of most number theory courses.
[[category:mathematics]]
[[category:mathematics]]