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I realize I am new to the community of online math forums, so I'm probably breaking a few etiquette rules (and possibly more important rules too - if so, please let me know and I'll fix what I can.)

However, I am working on a math problem, and I am stuck on bounding a particular equation, and I am looking for help.

I'm trying to bound a series of equations, each equation based on the first k primes greater than 3:

k = 1, f(x) = x - x*2/5

k = 2, f(x) = x - x*2/5 - x*2/7 + x*4/35

k = 3, f(x) = x - x*2/5 - x*2/7 - x*2/11 + x*4/35 + x*4/55 + x*4/77 - x*8/385

etc

The first wrinkle comes from the fact that x will always be an integer, and my answer must always be a positive integer.

For example, if x was 103, then in the first equation (-x*2/5) would seem to be -41.

However, the problem comes from the fact that the answers will not be completely evenly spread (we can't just use the floor or ceiling).

Specifically, if x was 103, then in the first equation (-x*2/5) could be -40, -41, or -42. The value changes for each value of x.

My concern comes from the third equation (and later equations, as we continue to get more and more terms in the equation as we continue adding prime numbers). Continuing the example, if x was 103, then:

-x*2/5 could be -40, -41, or -42. -x*2/7 could be -28, -29 or -30. -x*2/11 could be -18, -19 or -20. x*4/35 could be 8, 9, 10, 11 or 12. x*4/55 could be 4, 5, 6, 7 or 8. x*4/77 could be 4, 5, 6, 7 or 8. -x*8/385 could be 0, -1, -2, -3, -4, -5, -6, -7, or -8.

If I simply assume the largest possible answer for each one, then I end up in a situation where my answer is a negative integer (which I know it cannot be). So how do I put a bound on this equation such that I can ensure I'll always get a positive solution?

As an additional note, this equation is equivalent to (at least) one other equation, which is much easier to manipulate and put bounds on. However, since this is the original equation I am working with, I need to make sure any bounds have their basis within the original equation, and then transfer them to the other equation, in order to actually finish the problem.

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# Bounding a prime number based equation

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