For given positive integers a,b,c consider the set S = {a + bk + cn : k = 0, 1, 2,...,n = 0, 1, 2,...}. Characterize all triples a,b,c when S contains

I'll prove that for every natural , n>1, for given positive integers a₁, a₂, .....,a_n and S={x₁a₁+x₂a₂+......+x_na_n: a₁, a₂, ....,a_n are positive integers} S contains infinitely many primes if and only if no positive integer divides all n integers (a₁, a₂,......,a_n)

proof:

suppose there is a positive integer P, so that a_i is divisible by P, for every 0 1a ₁+x ₂a ₂+......+x _na _n is always divisible by P, and because there arent infinitely many primes that are divisible by any positive integer P (unless P=1, but we know P isnt 1) there arent infinitely many primes in S.

I'll prove that for every natural , n>1, for given positive integers a₁, a₂, .....,a_n and S={x₁a₁+x₂a₂+......+x_na_n: a₁, a₂, ....,a_n are positive integers}, there exists a positive integer D, so that for every D 1, x ₂, x ₃,......,x _n so that  x ₁a ₁+x ₂a ₂+......+x _na _n=y if no positive integer divides all n integers (a ₁, a ₂,......,a _n).

proof:

The proof is by induction- Suppose the positive integer P divides all positive integers a₁, a₂, .....,a_n-1, and that after dividing each of the integers by P no other integers divides them all. Lets define b_i=a_i/P for every 0 1 1, x ₂, x ₃,......,x _n-1 so that x ₁b ₁+x ₂b ₂+......+x _n-1b _n-1=y Therefore, for every y that is divisible by P and bigger than P*D ₁ there exists positive integers x ₁, x ₂, x ₃,......,x _n-1 so that  x ₁a ₁+x ₂a ₂+......+x _n-1a _n-1=y. Suppose a _n and P are coprime. Therefore, we can express every number of the form x ₁a ₁+x ₂a ₂+......+x _na _n using the sum r ₁*P+r ₂*a _n. Using the chicken-nugget theorem (Search it up, its real) we can infer that there exists an integer D ₂ so that for every D ₂ 1 and r ₂ so that r ₁*P+r ₂*a _n=y.

(If a_n and P are NOT coprime that means another positive integer, F, divides both of them, and that means F divides all of the integers (a₁, a₂,......,a_n), but that is a contradiction to what we said in the beginning- no positive integer divides all of the integers).

And therefore, if the theorem

for every natural , n>1, for given positive integers a₁, a₂, .....,a_n and S={x₁a₁+x₂a₂+......+x_na_n: a₁, a₂, ....,a_n are positive integers}, there exists a positive integer D, so that for every D 1, x ₂, x ₃,......,x _n so that  x ₁a ₁+x ₂a ₂+......+x _na _n=y if no positive integer divides all n integers (a ₁, a ₂,......,a _n).

Is true for n-1, it is also true for n.

I'll complete the theorem by proving it for n=2: suppose a₁ and a₂ are positive integers. if no positive integers divides both of them, then they are coprime integers, and using the chicken nugget theorem we can infer there exists a positive integer D₁ so that for every D₁ 1a ₁+x ₂a ₂=y for some positive integers x ₁ and x ₂.

Therefore, the theorem is true for n>1.

We know there are infinitely many primes, and we know that for every positive integer D there exists a FINITE number of primes between 1 and D, So there are infinitely many primes that are equal or bigger than D.  from the following theorem, we can infer Every prime that is bigger than D will be in the set S, and therefore, there are infinitely many primes in S IF AND ONLY IF no positive integer divides a₁, a₂,......,a_n.

Now, this was just a generalization of your question, and I'll use it to answer your question-

Suppose b and c are coprime. Therefore, there exists a positive integer D so that for every D 1b+x ₂c=y for some positive integers x ₁ and x ₂. therefore, for every a+D 1b+x ₂c=y for some positive integers x ₁ and x ₂. That means there are infinitely many primes in S.

so if b and c are coprime, we can infer that there are infinitely many primes in S. We also know that if there exists a positive integer P so that it divides a, b and c There is a finite number of primes in S.

The question is: what if there exists a positive integer P so that P divides b, and c, but not a? that is equivalent to the question: Is there an arithmetic sequence a_n=a+bn where a and b are coprime and that contains a finite number of primes?

unfortunately i cant answer that question, BUT i have this cool generalization (?) of the question, and i love generalizations, so i'll keep it although its useless.

EDIT: The answer to the question "Is there an arithmetic sequence a_n=a+bn where a and b are coprime and that contains a finite number of primes?" is NO, and therefore S contains infinitely many primes only and only if no positive integer divides both a, b, and c. But We have to prove it using Dirichlet's theorem on arithmetic progressions, So i cant really prove it.

Who gave you this question? because whoever did must be a horrible person.