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Back to back challenges I guess. Hopefully this one is more clear than the last.

Background

I was playing around with Goldbach's conjecture, and decided to plot evens (on the x-axis) vs the number of ways that two primes could be found to sum to a given even on the y axis.

The two plots I've made below in Python are for 10,000 and 1,000,000 evens.

enter image description here enter image description here

The structure of these graphs is really fascinating! So I decided to go further and color different types of evens (those divisible by different numbers) different colors.

What I found was that each band represented a set of evens divisible by some odd number, e.g., 3, 5, 7, 9. So that got me thinking of the following challenge. I would post the picture, but doing so might make this challenge a bit too easy, so I'll refrain till after I've crowned someone a victor.

Challenge

Define G(x) to be a function defined on the positive evens which returns the number of different ways to sum two primes numbers to form x. Note that the primes are indistinguishable, i.e., 3+5 and 5+3 are considered equivalent.

This is a challenge where your goal is the following:

Find 3 differentiable and concave functions f,g,h defined on the positive reals such that:

  1. For any even divisible by 6 and less than 10,000,000, and any real y less than or equal to this even, f(y)<=G(even)
  2. For any even divisible by 10 and less than or equal to 10,000,000, and any real y less than or equal to this even, g(y)<=G(even)
  3. For any even divisible by 14 and less than 10,000,000, and any real y less than or equal to this even, h(y)<=G(even)

Your score will be the integral from 0 to 10,000,000 of (f+g+h).

Rules

  • Each submission must not only include the formulas for all three of these functions, but show through code in whatever language they see fit that the conditions of the challenge are met.
  • Each submission must also calculate the integral of their score. This can be done in a separate language if so desired by the user (or by hand if you are a masochist).
  • Once a submission has been made, any submission whose score is not at least 5 points higher than the previous submission will not dethrone the current winning submission.

Prizes

I don't have much reputation, but I'll put up a +50 bounty for first place, along with a +15 for accept. This is somewhat of a new type of challenge for this site, so enjoy!

Edit

Considering the poor reception this problem has received, I will no longer be offering a prize for this problem. However, I'm working on designing another one, which hopefully will have a prize associated with it!

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  • 3
    \$\begingroup\$ It looks like you have also posted this challenge in the sandbox. I recommend deleting this one until you become confident that you can post it. \$\endgroup\$ – Erik the Outgolfer Aug 10 '18 at 20:42
  • \$\begingroup\$ for G(x) does 3+5 and 5+3 mean different way or are they the same? \$\endgroup\$ – Kroppeb Aug 11 '18 at 20:50
  • \$\begingroup\$ @Kroppeb Sorry, those are not different. I will add that to the question. \$\endgroup\$ – Don Thousand Aug 11 '18 at 21:05
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    \$\begingroup\$ This challenge is flawed because the score can be arbitrarily high and irrelevant to the problem. Functions \$f, g, h\$ are always zero for integer \$x\$: $$f(x) = g(x) = h(x) = c(1 - \cos(2\pi x))$$ But can have arbitrarily high integral. \$\endgroup\$ – orlp Aug 12 '18 at 10:48
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    \$\begingroup\$ On another note, I do not feel like "let[ting] this question die" is appropriate. Please either delete this question or edit it so that it is not left to die. \$\endgroup\$ – Jonathan Frech Aug 15 '18 at 7:13

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