On the Universal Encoding Optimality of Primes

The factorial-additive optimality of primes, i.e., that the sum of prime factors is always minimum, implies that prime numbers are a solution to an integer linear programming (ILP) encoding optimization problem. The summative optimality of primes follows from Goldbach’s conjecture, and is viewed as an upper efficiency limit for encoding any integer with the fewest possible additions. A consequence of the above is that primes optimally encode—multiplicatively and additively—all integers. Thus, the set P of primes is the unique, irreducible subset of ℤ—in cardinality and values—that optimally encodes all numbers in ℤ, in a factorial and summative sense. Based on these dual irreducibility/optimality properties of P, we conclude that primes are characterized by a universal “quantum type” encoding optimality that also extends to non-integers.

Deep learning-based approximation of Goldbach partition function

Goldbach’s conjecture is one of the oldest and famous unproved problems in number theory. Using a deep learning model, we obtain an approximation of the Goldbach partition function, which counts the number of ways of representing an even number greater than 4 as a sum of two primes. We use residues of number modulo first 25 primes as features and archive a more accurate approximation, which reduces error rate from [Formula: see text] to [Formula: see text].

Proof of the Goldbach's Conjecture

Goldbach’s Conjecture states that every even number greater than 3, can be written as a summation of two prime numbers. This conjecture is roughly 300 years old and a very famous unsolved mathematics problem. To prove the Goldbach’s Conjecture, I use the contradiction method in mathematics as below.

Arithmetic, enumerative induction and size bias

Number theory abounds with conjectures asserting that every natural number has some arithmetic property. An example is Goldbach’s Conjecture, which states that every even number greater than 2 is the sum of two primes. Enumerative inductive evidence for such conjectures usually consists of small cases. In the absence of supporting reasons, mathematicians mistrust such evidence for arithmetical generalisations, more so than most other forms of non-deductive evidence. Some philosophers have also expressed scepticism about the value of enumerative inductive evidence in arithmetic. But why? Perhaps the best argument is that known instances of an arithmetical conjecture are almost always small: they appear at the start of the natural number sequence. Evidence of this kind consequently suffers from size bias. My essay shows that this sort of scepticism comes in many different flavours, raises some challenges for them all, and explores their respective responses.

On the Distribution of the Nontrivial Zeros for the Dirichlet L-Functions

This paper addresses a variant of the product for the Dirichlet $L$--functions. We propose a completely detailed proof for the truth of the generalized Riemann conjecture for the Dirichlet $L$--functions, which states that the real part of the nontrivial zeros is $1/2$. The Wang and Hardy--Littlewood theorems are also discussed with removing the need for it. The results are applicable to show the truth of the Goldbach's conjecture.

On Primes of the Form 2x-q (where q is a prime less than or equal to x) and the Product of the Distinct Prime Divisors of an Integer (Revised): A Function Approach to Proving the Goldbach Conjecture by Mathematical Induction

This paper is really an attempt to solve the age-old problem of the Goldbach Conjecture, by restating it in terms of primes of the form 2x-q (where q is a prime less than or equal to x). Restating the problem merely requires us to ask the question: Does a prime of form 2x-q lie in the interval [x, 2x]? We begin by introducing the product, m, of numbers of the form 2x-q. Using the geometric series, an upper bound is estimated for the function m. Next, we prove a theorem that states every even number, 2x, that violates Goldbach’s Conjecture must satisfy an inequality involving a simple multiplicative function defined as the product, ρ(m), of the distinct prime divisors of m. A proof of the Goldbach Conjecture is then evident by contradiction as a corollary to the proof of the inequality.

Exploring Goldbach’s conjecture, the Pebonacci sequence and its five-dimensional vortex structure based on the logarithm of the circle

A method based on circle logarithm to prove Goldbach’s conjecture and Pebonacci sequence is proposed. Its essence is to deal with the real infinite series, each of the finite three elements (prime numbers, number series) has asymmetry problems, forming a basic even function one-variable quadratic equation and odd function one-variable three-dimensional number sequence; it is converted to "The irrelevant mathematical model expands latently in a closed interval of 0 to 1," forming a five-dimensional vortex space structure.

The Complexity of Mathematics

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true or this has an infinite number of counterexamples using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

The Complexity of Mathematics

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.