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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
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TWIN PRIME THEOREM
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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
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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
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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
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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}⇒⊤ ∎
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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′
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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 ,ℙ
ℙ
..., ℙ
1.1.
Next, Euclid created a theorem that showed that despite the enormous size implied by ℙ
an
additional, larger prime number exists by first determining the hypothetical product of all the
known prime numbers (ℙ′
):
Theorem: Despite the enormous size implied by ℙ
an additional, larger prime number
exists, since the product of the set of all prime numbers ℙ′
composite. If ℙ′
range. If ℙ′
within ℙ′
+ 1 must be prime or
+ 1 is prime then one new prime number exists beyond the defined
+ 1 is not prime, then it must be divisible by a prime number not contained
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
ℙ′
8
ℙ′ x ℙ′
ℙ′
..., ℙ′
ℙ′ x ℙ′
ℙ′
..., ℙ′
+1
ℙ′ x ℙ′
ℙ′
..., ℙ′
+ 1 mod ℙ′ = 1
1.2.
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 )
exist and by proxy a finite number of twin primes (ℙ
) exist, resolving the Twin Prime
Theorem with a contradiction (see equation 1.3).
≠ℙ ,ℙ
∴
≠ℙ ,ℙ
ℙ
..., ℙ
ℙ
1.3.
..., ℙ
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
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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.