Integer equations: find numbers not hit by 11x+4, 12x+11, 17x+12, 19x+1

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lukem

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Nov 10, 2017
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Hello

I am interested in making a set of equations more useful, but I can't seem to think of/remember if there is a way.

I want to find the numbers not hit by a set of equations.
The equations are simple, but I don't see a way to solve the equations, only comparing results

For instance

11x+4
12x+11
17x+12
19x+1

--2,3,5 through 10,13,14 aren't hit.

Im not interested in whether they cross paths on a graph, only the integer values they produce.

Is there a way to solve for the integer values that aren't reached? Or perhaps just check one integer for all the equations at once. I know its simple now, but if I have dozens or hundreds of these, I'd like to be able to simplify.

Thanks!
 
Plot these expressions in x as values of y

Plot each line.
Plot a lattice of dots at each integer grid intersection.

Any lattice dot without a line through it is a "missed value" in y.
 
Bob Brown MSEE said:
Plot each line.
Plot a lattice of dots at each integer grid intersection.

Any lattice dot without a line through it is a "missed value" in y.
Click to expand...


Thank you. I'm glad you see what I'm trying to do here, as I wasn't sure how to word the question. I have used that briefly, and then a table of values for each one, but I was hoping for some kind of hint of where to look, or if there was some way to find a guaranteed missed value without the manual labor.
 
lukem said:
Hello

I am interested in making a set of equations more useful, but I can't seem to think of/remember if there is a way.

I want to find the numbers not hit by a set of equations.
The equations are simple, but I don't see a way to solve the equations, only comparing results

For instance

11x+4
12x+11
17x+12
19x+1

--2,3,5 through 10,13,14 aren't hit.

Im not interested in whether they cross paths on a graph, only the integer values they produce.

Is there a way to solve for the integer values that aren't reached? Or perhaps just check one integer for all the equations at once. I know its simple now, but if I have dozens or hundreds of these, I'd like to be able to simplify.

Thanks!
Click to expand...

It's not quite clear what the goal is. I believe you want to find integer values n that are not attained by any of the functions

n = 11x + 4
n =12x + 11
n =17x + 12
n =19x + 1

for any non-negative integer value of x. But it is impossible to find ALL such n, since there are infinitely many of them; and if you want to find, say, all that are less than a given value, no single formula could do that -- it has to be an algorithm. (I would just go through an array of values, marking those that are attained, in the style of the Sieve of Eratosthenes.)

But if you just want to easily determine whether ONE value n is attained, that is easy enough. Solve each equation for x,

x = (n - 4)/11
x = (n - 11)/12
x = (n - 12)/17
x = (n - 1)/19

and find whether any of them give an integer value. (I'd use a mod function rather than division if you are writing a program.) If any do, then it is attained; if none do, it is not. I doubt that there is any shortcut beyond this.
 
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