Question

Number theory question: Is there a general way of finding all integer n for which \[\frac{\prod_{i=1}^n \left( a_i + b_i n \right)}{c + d n}\] is an integer (a,b,c,d are also integers)? I understand there may be an infinite set of possible n, but is there a way of finding the pattern of the n simply?

Answer 1

fix your tex I can't read your question

Answer 2

It works perfectly for me.

Answer 3

@henpen What are the limits for Pi?

Answer 4

It's a finite product of terms.

Answer 5

@genius12 lol

Answer 6

@Jack175 That's what it looks like and that's what I call it....makes sense no? lol

Answer 7

@henpen I know what the notation means lol I was asking you for the limits like in the above example it's k = 1 and goes to 5 etc...

Answer 8

Arbitrary, k=1 to some other integer. take 2 for simplicity if you want for the first solution

Answer 9

but k has to start at 1 or can it start at any integer? @henpen

Answer 10

It doesn't matter, call it 1.

Answer 11

Ok and what is the small subscript 'i' to the left of \(\bf a_i+b_in\)? @henpen

Answer 12

\[_{i=1}^n\{a_i\}\] is just a set of integers.

Answer 13

*

Answer 14

@henpen omg lol, you see there is a small subscript 'i' out side the numerator bracket in the corner near the Pi? What is that there for is what I'm asking rofl...

Answer 15

It's multiplying over all i in the set {1,...,n}.

Answer 16

It's shorthand for \[\prod_{i=1}^n\]

Answer 17

Are we to assume that the g.c.d (a,b,c,d) = 1?

Answer 18

@henpen I got a solution if you're still here. Also verify any conditions/restrictions that this question poses before I post the solution.

Answer 19

I'm here, and @domu no.

Answer 20

@genius12 ?

Answer 21

@henpen hold on a sec. answering a question.

Answer 22

@henpen K I am back. You there?

Answer 23

Yes

Answer 24

Ok I am going to make two assumptions here. Firstly, both {a} and {b} is the set all integers and that the i'th term in each set is the same hence \(\bf a_i=b_i\)

You ok with that? @henpen

Answer 25

No

Answer 26

I'll give an example.

Answer 27

Find all n such that \[\frac{(2n+3)(2n+25)}{5n+56}\]
is an integer

Answer 28

@henpen What does your solution look like to the problem you just posted?

Answer 29

It looks like an infinite set of possible n. I'm asking whether there's a way of generating that set without trying each one out.

Answer 30

Are the initial set of integers {a_i}, {b_i}, c, and d fixed or are we trying to find such a set for which the product will be an integer? Also, are each of the integers strictly positive?

Answer 31

I would like to know that as well. Are sets of {a} and {b} fixed or not. If they are not fixed then I don't know how one would solve this question. @henpen @domu