scholarly journals The Complexity of Number Theory

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Infinite Number ◽  
Riemann Hypothesis ◽  
Unsolved Problem ◽  
Short Proof ◽  
History Of Mathematics ◽  
Difficult Problem ◽  
Pure Mathematics ◽  
History Of ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

The Goldbach's conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach's conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A principal complexity class is NSPACE(S(n)) for some S(n). We show if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). This proof is based on the assumption that if some language belongs to NSPACE(S(n)), then the unary version of that language belongs to NSPACE(S(log n)) and vice versa. However, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. Since Harald Helfgott proved that the weak Goldbach's conjecture is true, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples, where the case of infinite number of counterexamples statistically seems to be unlikely. In addition, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Twin prime conjecture is true. Moreover, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Beal's conjecture is true. Since the Beal's conjecture is a generalization of Fermat's Last Theorem, then this is also a simple and short proof for that Theorem. In mathematics, the Riemann hypothesis is consider to be the most important unsolved problem in pure mathematics. If PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Riemann hypothesis is true.

scholarly journals The Complexity of Number Theory

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Number Theory ◽  
Infinite Number ◽  
Short Proof ◽  
History Of Mathematics ◽  
Difficult Problem ◽  
Fundamental Question ◽  
Complexity Classes ◽  
History Of ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

The Goldbach's conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach's conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A major complexity classes are L, NL and NSPACE(S(n)) for some S(n). Whether L = NL is a fundamental question that it is as important as it is unresolved. We show if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). This proof is based on the assumption that if some language belongs to NSPACE(S(n)), then the unary version of that language belongs to NSPACE(S(log n)) and vice versa. However, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. Since Harald Helfgott proved that the weak Goldbach's conjecture is true, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples, where the case of infinite number of counterexamples statistically seems to be unlikely. In addition, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Beal's conjecture is true when L = NL. On November 2019, Frank Vega proves that L = NP which also implies that L = NL. In this way, the Beal's conjecture is true and since the Beal's conjecture is a generalization of Fermat's Last Theorem, then this is also a simple and short proof for that Theorem.

scholarly journals The Complexity of Number Theory

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Number Theory ◽  
Infinite Number ◽  
Short Proof ◽  
History Of Mathematics ◽  
Difficult Problem ◽  
Complexity Class ◽  
Fermat's Last Theorem ◽  
History Of ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

The Goldbach's conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach's conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A major complexity class is NSPACE(S(n)) for some S(n). We show if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). However, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. Since Harald Helfgott proved that the weak Goldbach's conjecture is true, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples, where the case of infinite number of counterexamples statistically seems to be unlikely. In addition, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Beal's conjecture is true. Since the Beal's conjecture is a generalization of Fermat's Last Theorem, then this is also a simple and short proof for that Theorem.

scholarly journals The Complexity of Mathematics

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Number Theory ◽  
Complexity Theory ◽  
Riemann Hypothesis ◽  
Open Problems ◽  
Mathematical Proofs ◽  
Pure Mathematics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

The Complexity of Mathematics

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Number Theory ◽  
Complexity Theory ◽  
Riemann Hypothesis ◽  
Open Problems ◽  
Mathematical Proofs ◽  
Pure Mathematics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

The Complexity of Mathematics

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Number Theory ◽  
Complexity Theory ◽  
Riemann Hypothesis ◽  
Open Problems ◽  
Mathematical Proofs ◽  
Pure Mathematics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true or this has an infinite number of counterexamples using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

scholarly journals The Complexity of Mathematics

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Infinite Number ◽  
Riemann Hypothesis ◽  
Complexity Class ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

The strong Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two primes. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A principal complexity class is NSPACE(S(n)) for some S(n). We show if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). However, if this happens, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. In addition, if this happens, then the Twin prime conjecture is true. Moreover, if this happens, then the Beal's conjecture is true. Furthermore, if this happens, then the Riemann hypothesis is true. Since the weak Goldbach's conjecture was proven, then this will certainly happen.

scholarly journals The Riemann Hypothesis is True

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Riemann Hypothesis ◽  
Unsolved Problem ◽  
Natural Numbers ◽  
Chebyshev Function ◽  
Pure Mathematics ◽  
Sum Of Divisors

The Riemann hypothesis has been considered the most important unsolved problem in pure mathematics. The David Hilbert's list of 23 unsolved problems contains the Riemann hypothesis. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is true.

scholarly journals The Riemann Hypothesis is most likely true

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Riemann Hypothesis ◽  
Unsolved Problem ◽  
Natural Numbers ◽  
Chebyshev Function ◽  
Pure Mathematics ◽  
Sum Of Divisors

The Riemann hypothesis has been considered the most important unsolved problem in pure mathematics. The David Hilbert's list of 23 unsolved problems contains the Riemann hypothesis. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is most likely true.

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