#### Title

The Twin Prime-Goldbach Conjecture

#### School Name

South Carolina Governor's School for Science & Mathematics

Mathematics

Mentored

#### Abstract

The Twin Prime Conjecture states that there are infinitely many pairs of primes differing by two. The Goldbach Conjecture states that every even integer greater than two is the sum of two primes. In this research, we investigate a combined version of these two conjectures which we call the Twin Primes-Goldbach Conjecture. It is conjectured that every sufficiently large multiple of 6 can be expressed as p+q and (p+2)+(q-2) where p, p+2 and q-2, q are pairs of twin primes. There are clear patterns in the plots of the number of Goldbach representations of even integers. Hardy and Littlewood introduced a correction factor to explain these patterns. We introduce an analogous correction factor called tpg(n) for the Twin Prime-Goldbach problem. We present numerical evidence suggesting that the number of Twin Prime-Goldbach representations behaves like a random normal distribution centered around C, tpg(n),n/log(n)^4. A series of statistical analyses is used to determine how well the representations model an approximate normal distribution.

Furman Hall 121

#### Start Date

3-28-2020 12:15 PM

Oral Only

#### Group Project

No

COinS

Mar 28th, 12:15 PM

The Twin Prime-Goldbach Conjecture

Furman Hall 121

The Twin Prime Conjecture states that there are infinitely many pairs of primes differing by two. The Goldbach Conjecture states that every even integer greater than two is the sum of two primes. In this research, we investigate a combined version of these two conjectures which we call the Twin Primes-Goldbach Conjecture. It is conjectured that every sufficiently large multiple of 6 can be expressed as p+q and (p+2)+(q-2) where p, p+2 and q-2, q are pairs of twin primes. There are clear patterns in the plots of the number of Goldbach representations of even integers. Hardy and Littlewood introduced a correction factor to explain these patterns. We introduce an analogous correction factor called tpg(n) for the Twin Prime-Goldbach problem. We present numerical evidence suggesting that the number of Twin Prime-Goldbach representations behaves like a random normal distribution centered around C, tpg(n),n/log(n)^4. A series of statistical analyses is used to determine how well the representations model an approximate normal distribution.