# Bounding a prime number based equation

by Nelphine
Tags: based, bounding, equation, number, prime
 P: 11 Hello all! 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.
P: 891
 Quote by Nelphine Hello all! 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.
I can't make sense of the statement that if x = 103 then (-x*2)/5 could be -40,-41, or 42 since 103*2/5 = 41.2. If you explain the problem with more detail to avoid confusion, you stand a better chance of getting an answer to the question.
P: 891

## Bounding a prime number based equation

You are looking on a way to determine a more restricted bound on {m,n,o,p,...}. Lets call them m(i). As far as I understand m(i) appear in the equations f(x) = y - Sum[ for 3< prime(p) < p(n), 2^j *[Floor[y/ Product(p(1),P(2)...(p(j))] + m(i)]. As far as I understand 0<= m(i)<= 2^j. You attempt to show how the occasion of bad results can limit the value for "o" to > 1 in the third to the last paragraph of your last post, but I cant see how to determine good results from bad results or whether the result for m = 0,n = 2, o = 0 is any different from the result for m = 0, n = 0, o = 2. Accordingly, I need a better understanding of your problem to help you.
 P: 11 My attempt at a proper formula, although I know it's not quite right: given k, let m = [p(k)^2 - p(k)]/6, where p(k) = the kth prime number, then: f(k) = m + sum [ from i = 1 to k, sum ( from j = 3 to k, (-1)^i * 2^i *floor[m/ product ( from L = j to k, p(L) )] +n(i,j) ) ] where n(i,j) is an element from the set {0,1,...,2^i} I think there's still something wrong with the product part of it (specifically what L should range over), sigh.. (L should range over i elements, but how do I formalize which elements??? (since for instance specific terms could be p(3)*p(9) if i = 2, or it could be p(4)*p(6)*p(7) if i = 3.)
 P: 11 Another thing that has been pointed out to me that is wrong with that formula: I forgot to include a set of brackets containing [2^i *floor[m/ product ( from L = j to k, p(L) )] +n(i,j)], as both of those terms should be affected by the (-1)^i

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