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Running head: TWIN PRIME THEOREM – COUNTEREXAMPLE Twin Prime Theorem – Counterexample Chris Gilbert Waltzek Northcentral University ©2014 Chris Waltzek, All Rights Reserved 1 TWIN PRIME THEOREM 2 Acknowledgements Special thanks to Professor Roger C. Tutterow (Mercer University), a mentor and friend, for invaluable encouragement and input on this and many other projects. In addition, thanks to mentor, Dr. Keshwani, for his passion for teaching and to NorthCentral University and professors, for providing a profound learning format. TWIN PRIME THEOREM 3 Abstract 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 a finite number of prime numbers and by proxy twin prime pairs exist resolving the Twin Prime Theorem with a contradiction. 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. Keywords: Twin Prime Theory, Clay Mathematics Institute, Euclid’s prime theorem, Goldbach’s Weak Conjecture TWIN PRIME THEOREM 4 Twin Prime Theorem – Counterexample Prime Number Conjecture 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 on the conjecture, revealing that the modern definition of a prime number may require revision. Theorem Premise #1: Assume that all prime numbers are the sum of 3 smaller primes, not only those > 5 (as proposed by Goldbach to Euler) with only one exception, the number 1 (1 was assumed prime at the time of Euler and Goldbach). 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? Given the 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: Proof 1. 3+1+1=5 1+1+1=3 1+1+?≠2 TWIN PRIME THEOREM 5 Quod Erat Demonstrandum (Q.E.D.). Therefore, the modern definition of a prime number is de facto, incomplete. 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. Proof 2. In the second proof, all numbers, including the number 1 are shown to include at least 2 out of 3 of the revised prime number criteria, except the number 2: 1. Odd number; 2. Only factors are 1 and itself, 3. Composed of the sum of 3 smaller primes. The numbers 5, 3, 2, and 1 are examined for primality: #5: 3 + 1 + 1 = 5; factors (5, 1); odd number (3 out of 3); #3: 1 + 1 + 1 = 3; factors (3, 1); odd number (3 out of 3); #2: 1 + 1 + ? = 2; factors (2, 1); even number (1 out of 3); #1: 1 + ? + ? = 1; factors (1, 1); odd number (2 out of 3); Deduction: Given that the number 2 is the only prime number that meets only 1 of the 3 prime number criteria, it is de facto not a prime number. Since the number 1 meets 2 out of 3 of the requirements, it is proposed to be a weak prime. In addition, this creates a new prime twin set (1, 3) adding further proof that 2 is not prime, since every twin prime must be separated by an even and therefore composite number. A logical proof follows (see Equation 1.1): TWIN PRIME THEOREM ∴{x | x ∈ ℙ′, 1 ⊂ x ∧ 2 ⊄ x}⇒⊤ ∎ 6 1.1. Hóper édei deîxai (OE∆) The findings suggest that the axioms underpinning prime numbers require adjustment: (2 ≠ ℙ′), and (1 = ℙ′). Although the proof implies that no even primes exist, in the event that a large even prime is eventually identified, the Prime Number Conjecture proof remains legitimate as long as the new prime is composed of the sum of three smaller primes. The new prime would be classified as a weak prime, like the number one, meeting 2 of the 3 prime criteria. New Twin Prime Using the axioms underpinning the Prime Number Conjecture, a new and unrecognized twin prime emerges: Theorem Given the Prime Number Conjecture and the twin prime formula (n = ℙ′); n + 2 = twin ℙ′′), a previously unknown twin number set includes 1 and 3: Proof: n = ℙ′; n + 2 = ℙ′′ 1 = ℙ′; 1 + 2 = ℙ′′ 1 = ℙ′; 1 + 2 = 3 = ℙ′′ 1, 3 = ℙ′′ Alternative Proof Given a specific, positive, even, and natural number, an alternative proof yields similar results: (2ℤ): n = 2ℤ; n +/- 1 = ℙ′′ 2 + 1 = 3 = ℙ′ TWIN PRIME THEOREM 7 2 - 1 = 1 = ℙ′ 1, 3 = ℙ′′ Deduction Given the Prime Number Conjecture and the twin prime formula, a previously undiscovered twin prime set emerges, 1, 3 = ℙ′′. The potential significance of this proof is in the field of number theory, more specifically in the search for proof or a counterexample to the Twin Prime Theory, for which the Clay Mathematics Institute offers a $1,000,000, Millennium Prize. Twin Prime Theorem According to Williamson (1782), Euclid included a proof showing that the number of prime numbers (ℙ′) is infinite (∞) in the magnum opus Elements. First, Euclid defined a finite list of prime numbers (see Expression 1.1): ℙ′1 , ℙ′2 ℙ′2 , ..., ℙ′𝑛 1.1. Next, Euclid created a theorem that showed that despite the enormous size implied by ℙ′𝑛 an additional, larger prime number exists by first determining the hypothetical product of all the known prime numbers (ℙ′𝑝𝑟𝑜𝑑 ): Theorem: Despite the enormous size implied by ℙ′𝑛 an additional, larger prime number exists, since the product of the set of all prime numbers ℙ′𝑝𝑟𝑜𝑑 + 1 must be prime or composite. If ℙ′𝑝𝑟𝑜𝑑 + 1 is prime then one new prime number exists beyond the defined range. If ℙ′𝑝𝑟𝑜𝑑 + 1 is not prime, then it must be divisible by a prime number not contained within ℙ′𝑝𝑟𝑜𝑑 because the remainder is always 1 which cannot be divided by a prime number, resulting in a logical contradiction. Therefore, the original hypothesis that there is a finite number or primes is false; ergo there exist an infinite number of primes (see equation 1.2). Proof (see equation 1.2) TWIN PRIME THEOREM 8 ℙ′prod = ℙ′1 x ℙ′2 x ℙ′2 x, ..., x ℙ′𝑛 1.2. ℙ′1 x ℙ′2 x ℙ′2 x, ..., x ℙ′𝑛 + 1 ℙ′1 x ℙ′2 x ℙ′2 x, ..., x ℙ′𝑛 + 1 mod ℙ′𝑛 = 1 Therefore, Euclid proved that for any finite list or prime numbers, there exists a larger prime number on the list. However, if 1 is found to be divisible by a prime number, then the proof is false and a finite number of primes exist. Since the theorems of this paper suggest that 1 is a prime number, Euclid’s proof is found to be false. Thus, a finite number of prime numbers (ℙ′1 ) exist and by proxy a finite number of twin primes (ℙ′′𝑛 ) exist, resolving the Twin Prime Theorem with a contradiction (see equation 1.3). lim 𝑓(n) ≠ ℙ′1 , ℙ′2 ℙ′2 , ..., ℙ′′𝑛 1.3. 𝑛→∞ ∴ lim 𝑓(n) ≠ ℙ′′1 , ℙ′′2 ℙ′′2 , ..., ℙ′′𝑛 𝑛→∞ 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 a finite number of prime numbers and by proxy twin prime pairs exist resolving the Twin Prime Theorem with a contradiction. 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. Williamson, J. (1782). The elements of Euclid, with dissertations. Clarendon Press, Oxford.