Twin Prime Theorem 3

Running head: TWIN PRIME THEOREM 1

Twin Prime Conjecture

Chris Gilbert Waltzek

Northcentral University

Acknowledgements Special thanks to mentor, Dr. Keshwani, for his passion for teaching and to NorthCentral

University and professors, for providing a profound learning format.

Abstract TWIN PRIME THEOREM 3

This paper builds on Goldbach’s weak conjecture, showing that all primes to infinity are composed of 3 smaller primes, suggesting that the modern definition of a prime number may be incomplete and requires revision. The results indicate that the axioms underpinning prime numbers should include one as a prime number and two as a non-prime number, adding a new dimension to the most fundamental of all integers. Given the interpolated definition of a prime number, en passant, results with a unique solution to the Twin Prime Conjecture. Euclid’s Prime

Theorem combined with the deductions of Theorem I, illustrate that an infinite number of prime pairs exist. The potential significance to this proof is in the field of number theory, more specifically in the search for a proof of the Twin Prime Theory, for the Clay Mathematics

Institute.

Twin Prime Theorem In a letter to the great mathematician Leonard Euler, Goldbach posited that all prime numbers

(ℙ′) greater than five (ℙ′ > 5) are the sum of 3 smaller primes, known as Goldbach’s weak conjecture (Bruckman (2006); Bruckman (2008); Chang (2013) & Shu-Ping (2013). The conjecture was recently proven true therefore; all primes to infinity are composed of 3 smaller primes. The following theorem builds upon the conjecture to show that the modern definition of a prime number may require revision. In addition, a second theorem builds upon the first as a proposed solution to the Twin Prime Conjecture.

Theorem I

Premise #1: Assume that all prime numbers are the sum of 3 smaller primes and not just those >

5 (as proposed by Goldbach to Euler) with only one exception, the number 1 (1 was assumed to be prime at the time of Euler and Gold

Premise #2: Assume that the number two is not, prime. This claim is intuitive, not one even prime has been identified for any number up to 17 million digits in length, so why should it be assumed that the even number 2 is prime?

Conversely, the empirical facts suggest that the current definition of a prime number requires revision. Given the two premises, a proof follows that resolves the problem in Goldbach's Weak

Conjecture regarding primes of less than 6, en passant proving the primality of 1 and non- primality of 2. Contrary to the work of Goldbach, the numbers 5 and 3 are shown to be composed of 3 smaller prime numbers and the number 2 does not fit the revised definition of a prime number.

3 + 1 + 1 = 5

1 + 1 + 1 = 3 TWIN PRIME THEOREM 5

1 + 1 + ? ≠ 2

Quod Erat Demonstrandum (Q.E.D.).

Therefore, the modern definition of a prime number is de facto, incomplete. The number 1 is prime and the number two is not prime. A new prime number definition follows:

Deduction

1. A prime number is an odd and natural number,

2. composed of the sum of three smaller prime numbers,

3. with only two factors: 1 and itself.

The only exception is 1, which is presumed to be a special case prime, almost transcendental (π =

3.14, e = 2.71, & Φ = 1.618), the mother / father of all prime numbers and the only common factor of all prime numbers. A logical proof follows (see Equation 1.1):

∴{x | x ∈ ℙ′, 1 ⊂ x ∧ 2 ⊄ x}⇒⊤ ∎ 1.1.

Hóper édei deîxai (OE∆)

Although the results seem trivial on the surface, the findings suggest that the axioms underpinning prime numbers may require adjustment (2 ≠ ℙ′), adding a new dimension to the most fundamental of all integers, prime numbers.

Twin Prime Theorem Given the outcome of Theorem I (1 = ℙ′; 2 ≠ ℙ′) and the twin prime formula (n = ℙ′; n + 2 = twin ℙ′ (ℙ′′)), a previously unknown twin number set is recognized (ℙ′′ = 1, 3). The new twin set provides the support for the deduction that an infinite number of prime pairs exists:

Theorem II

Premise #1: Assume that Euclid’s prime number proof is correct; an indefinite number of primes exist.

Premise #2: Assume the outcome of Theorem I (1 = ℙ′; 2 ≠ ℙ′) and the twin prime formula (n = ℙ

′; n + 2 = twin ℙ′ (ℙ′′)), result with a previously unknown twin number set (ℙ′′ = 1, 3).

(n = ℙ′); n + 2 = ℙ′′)

1 = ℙ′; 1 + 2 = ℙ′′

1 = ℙ′; 1 + 2 = 3 = ℙ′′

1, 3 = ℙ′′

Alternative Proof

Given a specific, even, and natural number (2ℤ): TWIN PRIME THEOREM 7

(n = 2ℤ); n +/- 1 = twin (ℙ′′))

n = 2; 2 + 1 = 3 = ℙ′′

n = 2; 2 - 1 = 1 = ℙ′′

1, 3 = ℙ′′

Deduction

Given the Prime Number Conjecture and the twin prime formula, a previously undiscovered

prime set is derived (1, 3 = ℙ′′). Since the most basic even integer (2) is shown to be comprised

of two smaller prime numbers (1, 1), and there exists an infinite number of prime numbers as proven by Euclid’s Prime Theorem, it is proposed that the Prime Number Conjecture gains strength (see Equation 1.2):

{x | x ∈ 2ℤ, ℙ′ + 2 = }∎ 1.2. Discussion

This paper builds on Goldbach’s Weak Conjecture, showing that all primes to infinity are composed of three smaller primes, suggesting that the modern definition of a prime number may be incomplete and requires revision. The results indicate that the axioms underpinning prime numbers should include one as a prime number and two as a non-prime number, adding a new dimension to the most fundamental of all integers. In addition, Euclid’s Prime Theorem combined with the deductions of Theorem I, illustrate that an infinite number of prime pairs may exist. The potential significance to this proof is in the field of number theory, more specifically in the search for a proof of the Twin Prime Theory, for the Clay Mathematics Institute. TWIN PRIME THEOREM 9

References

Bruckman, P. S. (2006). A statistical argument for the weak twin primes conjecture. International

Journal Of Mathematical Education In Science & Technology, 37(7), 838-842.

Bruckman, P. (2008). A proof of the strong Goldbach conjecture. International Journal of

Mathematical Education In Science & Technology, 39(8), 1102-1109.

doi:10.1080/00207390802136560

Chang, Y. C. (2013). Layman's method to verify Goldbach's conjectures. World Journal of

Engineering, 10(4): 401-404. doi: 10.1260/1708-5284.10.4.401

Shu-Ping Sandie Han1, s. (2013). Additive number theory: Classical problems and the structure

of sumsets. Mathematics & Computer Education, 47(1), 48-58.