Following the example of the mathematician Paul Erdős, we have decided to give away prizes for the solution of our unsolved problems. These are some “open points” of our proof strategies, the ones that we consider most significant to get closer to the final goal, that is the proof of Goldbach’s conjecture:

ID Problem Prize Status
1 Complete in all details the proof of the Hypothesis H.1.T.A.2 (Explicit Hypothesis H.1.T.A for second order dashed lines) 20€ No solution received
2 Find a function f: \mathbb{N}^{\star} \Rightarrow \mathbb{N} such that, for each third order linear dashed line T = (n_1, n_2, n_3) and for each x \gt 0:

f(x) - (n_3)^2 \leq \mathrm{t\_space}_T(x) \leq f(x) + (n_3)^2

The function f must be expressed through a formula that only includes operators of sum, difference, product, division, integer part approximated by excess and defect, distinction between a finite number of cases.
As a starting point, the article Calculation of \mathrm{t\_space} for dashed lines of arbitrary order can be considered. With reference to the last open point, as the function \delta we chose (n_3)^2 because this (constant) function varies roughly as the width of the validity interval in dashed lines T_k, so on the basis of the solution you will send, a demonstration could be made of the existence of spaces within the validity range.

50€ No solution received
3 Find a generalization for the Propositions L.C.5 (Spaces of a third order dashed line preceding a dash on the first row) and L.C.6 (Spaces of a third order dashed line following a dash on the first row), considering the case of a linear dashed line of any order. 100€ No solution received

If you think you have found the solution for one of these problems:

  1. Write it down in all the details, in Italian or English, and send it to . You can use the format you prefer, as long as it is easily readable, for example Word with formulas written in MathType, PDF generated by LaTeX, or HTML pages that make use of the KaTeX or MathJax libraries; we prefer the HTML format, because it would simplify the publication on our site if the solution is accepted.
  2. We will reply in a short time confirming the receipt of your email. At the same time, we will update the status of the problem in this page, to make public the fact that we have received a solution.
  3. We will review your solution in order to check if it is valid; this phase, which is the most delicate, may take some time. If we have received multiple solutions for the same problem, we will examine them all in order of receipt. Only the first solution that we consider valid will be entitled to the prize, but all valid solutions can be published on our website.
  4. If we consider your solution invalid, for example because it is incorrect or incomplete, we will notify you, possibly proposing improvements, so that you can send us a new version. If you send it modified to us, we will examine it again, restarting from step 3.
  5. If we consider your solution valid, we will ask you for your consent to publish it on our site under the Creative Commons Attribution-ShareAlike 3.0 Unported License, quoting your name (or the pseudonym you prefer). This consent is mandatory for the purpose of receiving the prize.
  6. Once received the requested consent, we will update the status of the problem with the wording “Solved, to be published”
  7. We will ask you for your bank account details to proceed with the payment of the prize. You will receive a bank transfer from an Italian account, with reason for payment: “solution for problem X”, where X is the ID of the problem. At the moment there are no other ways to pay the prize.
  8. We will publish your solution on our website, respecting the Creative Commons Attribution-ShareAlike 3.0 Unported License.. At the same time, we will remove the problem from this page.