### Ideas for possible proofs

- Aldo Pappalepore, Investigation of Hardy-Littlewood and of Goldbach conjectures with the primality theorems of Congruence and of Complementary Congruence, 2023
- Aldo Pappalepore sent us a detailed study which, starting from the properties of congruences and using analytical techniques, traces a path for a possible proof of the Goldbach Conjecture (while addressing, in a similar way, the Twin Prime Conjecture). In lemma d) of Appendix B there is still an open point but, regardless of this aspect, the document is full of ideas and intermediate results potentially useful for the proof.
- Ultima, About the exceptions to Goldbach’s conjecture III, 2022
- Our attached reader Ultima continues to develop his theory about the exceptions to Goldbach’s Conjecture. In this work he introduces some hypotheses that let write even numbers into forms similar to Conjecture’s one, also using a known theorem by Wen Chao Lu which asymptotically bounds the number of possible exceptions. The document ends with some clarifications on what means to represent an integer number into a certain form, and about the content of the set “G2” used throughout the document.
- Michele Bertolino, Every power of 2 is the sum of two prime numbers, 2022
- Michele Bertolino sent to us this proof, the development of which is still to be completed, of a subcase of Goldbach’s Conjecture, which considers only powers of two. In spite of this limitation, the paper is very interesting for the originality of the methods, shown by means of some geometrical constructions, as well as for the presence of rather important intermediate results, like a connection between the Conjecture and the couples of twin primes (Theorem 4.1). The proof, though achieving the final goal, has one open point (Conjecture 5.2), which is still under analysis: is someone able to prove it?
- Ultima, About the exceptions to Goldbach’s conjecture II, 2021
- Our reader Ultima, after having supposed the existance of exceptions to Goldbach’s Conjecture (see also the other contribution of the same author listed below in this page), continues his argument attempting to find a form for expressing such exceptions, observing that some forms can be reduced to other ones. The final goal is to find a connection between Lemoine’s Conjecture and Goldbach’s weak Conjecture.
- Ultima, About the exceptions to Goldbach’s conjecture, 2020
- Is it possible to prove Goldbach’s Conjecture by contradiction, starting from the existence on a hypothetical set of exceptions? Our reader Ultima tries to sketch out an argument of this kind. The starting point is that every even number greater than 6 can be expressed as a sum of four prime numbers, which is a consequence of the weak Goldbach’s Conjecture, proved by Helfgott.
- Francesco Di Noto and Michele Nardelli, Theorem about the number of Goldbach’s pairs up to even N, 2019
- Discussion, with numerical examples, of some estimates for the number of Goldbach’s pairs. These estimates are based on some interesting correlations, for every even number N \geq 4, between the number of Goldbach’s pairs and the number of prime divisors.
- Francesco Di Noto and Michele Nardelli, Weak Goldbach’s conjecture already proved. The strong conjecture follows, 2016
- Proof sketch of the strong Goldbach’s Conjecture based on its weak version. Also some numerical evidences are discussed, for example about the number of Goldbach’s pairs (this topic will be studied in detail in a later paper by the same authors, see above). The paper ends with some mentions about Fermat’s factorization and Lagarias’ equivalent of the Riemann Hypothesis, RH1.

### Other contributions

- Ultima, Ultima – Almost all odd numbers can be understood as the sum of three prime numbers (alternative proof), 2023
- Our reader Ultima noted that, started from Wen Chao Lu’s theorem about the exceptions to Goldbach’s Conjecture, it’s possible to obtain a statement very similar to weak Goldbach’s Conjecture.
- Giovanni Di Savino, Goldbach’s conjecture satisfied with the strategy which youhg Gauss invents for summing numbers from 1 to 100, 2022
- Giovanni Di Savino offers us a wide-ranging reflection about Goldbach’s Conjecture, touching on different themes of modern and ancient mathematics, with numerous sitographic references.