Show that if each number {b, a + b, 2a + b, ...., (n-1)a + b} is prime then every prime p <= n, must divide a, i.e. p | a

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- May 18th 2010, 12:59 AMSmithPrime numbers.
Show that if each number {b, a + b, 2a + b, ...., (n-1)a + b} is prime then every prime p <= n, must divide a, i.e. p | a

- May 18th 2010, 05:54 AMtonio

Note that, then if is any prime dividing we'd have that__if we had__

.

But , and this is impossible since is a prime

and since there are different numbers in then there must exist , s.t.

any prime dividing also divides (hmmm**...why**?? Just a little explanation more...(Giggle)).

Tonio - May 18th 2010, 07:59 AMsimplependulum

We have for otherwise , that term would not be a prime .

Also because is not a prime .

Therefore ,

Let's consider the lemma ,

is divisible by provided that is prime to

Proof :

Take modulo and since is prime to it , there is an inverse of it , let

, it is deduced by using the fact that is an integer .

Consider

It should be divisible by if is prime to but we have each term is prime , we have the product contains prime factors that all are greater than , which is prime to . Therefore , is prime to is false for and we have that contains all prime .

Consider , ,

contains these two prime numbers but not also . It is not true that every prime , must divide a , it should be - May 18th 2010, 08:09 AMwonderboy1953Thought
These would be excellent questions for the math challenge section.

- May 18th 2010, 09:38 PMsimplependulum
I find that i can simplify the solution i gave ... Maybe the first thing is to modify the lemma i introduced ...

Lemma:

is divisible by where provided that is prime to .

The proof is quite similar to that of the original lemma .

Then , we just need to consider the product :

Let be the prime not exceeding ie .

It is divisible by if is prime to . However , each of the terms is prime so it doesn't contain the prime , a contradiction makes us to say ' is prime to ' is false and .

Note: We can also say for if . In the counterexample I gave , it is coincident that . Not being in this case , the statement is true .