ABSTRACT Numerical evidence suggests that for only about 2% of pairs p, p + 2 of twin primes, p + 2 has more primitive roots than does p.

Primitive Root Bias for Twin Primes

We show that there are only finitely many pairs of twin primes $$(p, p+2)$$(p,p+2) such that there exists an$$\mathcal{S}$$S-Diophantine quadruple in the sense of Szalay and Ziegler for the set$$\mathcal{S}$$S of integers composed only of primes p and p + 2.

Diophantine $$\mathcal{S}$$S-quadruples with two primes which are twin

We establish, via a formal/heuristic Fourier inversion calculation, that the Hardy-Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height E on the critical line.

Twin prime correlations from the pair correlation of Riemann zeros

We establish, via a formal/heuristic Fourier inversion calculation, that the Hardy-Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height $E$ on the critical line.

Twin prime correlations from the pair correlation of Riemann zeros

An analogous result for the twin prime conjecture is obtained by Ram Murty and Vatwani [13].

On the Connection Between the Goldbach Conjecture and the Elliott-Halberstam Conjecture

For example, if we denote the divisor function by τ , in 1952 Erdös and Mirsky [1] asked whether the equation τ(n) = τ(n+1) admits infinitely many solutions in the set of natural numbers, a question that can be considered as a close relative of the twin prime conjecture.

Multiplicative functions on shifted primes

An efficient numerical method of traversal of twin prime pairs was applied to verify bounds on the distribution of twin primes at an arbitrary distance.

Bounds in Twin Prime Distribution

Let p = 8k + 5, q = 8k + 3 be the twin prime pair for some nonnegative integer k.

Integral points on the elliptic curve Epq: y2 = x3 + (pq − 12) x − 2(pq − 8)

18) is that this formula enables correct calculations in set N on finding the multitude of twin prime numbers, in contrary of the above logarithmic relation which is an approximation and must tend to be correct as ν tends to infinity.

A Solution to the Famous “Twin’s Problem”

Twin prime numbers are two prime numbers which have the difference of 2 exactly.

Proof of Twin Prime Conjecture

10.3390/math7050400

, twin primes, prime triplets, etc.

Predicting maximal gaps in sets of primes

10.1007/S11537-019-1837-Z

The Twin Prime Conjecture asserts that there should be infinitely many pairs of primes which differ by 2.

The twin prime conjecture

10.1007/978-3-030-14799-0_60

Is the final output of the ATM can be produced at the halting state? We supported our analysis by reasoning on Thomson’s paradox and by looking closely the result of the Twin Prime conjecture.

Approximate Outputs of Accelerated Turing Machines Closest to Their Halting Point

10.1016/j.jnt.2019.08.003

Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies $\phi(p-1) \geq \phi(p+1)$ for a majority of twin-primes pairs $p,p+2$ and that the reverse inequality holds for a small positive proportion of the twin primes.

Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function

10.1007/S13226-019-0329-4

Let p = 8k + 5, q = 8k + 3 be the twin prime pair for some nonnegative integer k.

Integral points on the elliptic curve Epq: y2 = x3 + (pq − 12) x − 2(pq − 8)

10.12921/cmst.2019.0000033

In 2011 Wolf computed the "Skewes number" for twin primes, i.

On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood

10.1016/j.dam.2018.08.005

We find connections between prime product and prime power distance graphs and the Twin Prime Conjecture, the Green-Tao Theorem, and Fermat's Last Theorem.

Prime power and prime product distance graphs

10.1103/PhysRevLett.122.090201

Further, diffraction physics connections to number theory reveal how to encode all Gaussian primes, twin primes, and how to construct wave fields with amplitudes equal to the divisor function at integer spatial frequencies.

Simple Wave-Optical Superpositions as Prime Number Sieves.

10.4236/apm.2019.99038

18) is that this formula enables correct calculations in set N on finding the multitude of twin prime numbers, in contrary of the above logarithmic relation which is an approximation and must tend to be correct as ν tends to infinity.

A Solution to the Famous “Twin’s Problem”

10.1007/978-3-030-03661-4_2

These include maximum length binary signals based on shift register sequences, as well as several other classes of binary and near-binary signals, namely, the quadratic residue binary and ternary, Hall binary and twin prime binary signals.

Design of Pseudorandom Signals for Linear System Identification

10.3390/sym11060775

Moreover, an extension of this procedure to the case of twin primes is formulated.

An Investigation on the Prime and Twin Prime Number Functions by Periodical Binary Sequences and Symmetrical Runs in a Modified Sieve Procedure

10.32871/RMRJ1806.02.05

The highly irregular and rough fluctuations of the twin primes below or equal to a positive integer x  are considered in this study.

An Analytic Approximation to the Density of Twin Primes

10.15580/GJSETR.2019.1.040919068

At the same time, it is possible to guess the possibility of twin prime conjecture.

Even numbers are the sum of two prime numbers

10.20944/PREPRINTS201908.0207.V1

Twin prime numbers are two prime numbers which have the difference of 2 exactly.

Proof of Twin Prime Conjecture

10.1007/s00026-019-00469-0

For fixed odd integers $$t\ge 3$$, the sequence of E(d, t, x) with d running through the integers produces, conjecturally, sequences of “twin composites” analogous to the twin primes of the integers.

Twin Composites, Strange Continued Fractions, and a Transformation that Euler Missed (Twice)

10.11648/J.SI.20190702.11

Zhang Yitang from the University of New Hampshire offers the best research results of the infinity of twin primes.

Study on the Infinity of Twin Primes by Applying Sundaram’s Sieve Method

10.11648/J.IJAMTP.20190503.15

Although the mathematicians all over the world offered hard explorations of more than one hundred years, the proof of using pure mathematical theories on the conjecture of twin primes has not born in the world.

A Proof on the Conjecture of Twin Primes

Twin Prime
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