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 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):  
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 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.

Even Perfect Numbers—An Update

1981 ◽  
Vol 74 (6) ◽  
pp. 460-463
Author(s):  
Stanley J. Bezuszka
Keyword(s):  
Number Theory ◽  
Problem Solving ◽  
History Of Mathematics ◽  
Independent Study ◽  
History Of ◽  
Perfect Numbers ◽  
Excellent Resource

Do you have students who are computer buffs, always looking for a new problem to program efficiently? Do you have students who do independent study projects? If so, motivate them with this topic that is rich in the history of mathematics and number theory—perfect numbers. They provide an excellent resource for theoretical as well as computerized problem solving.

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.

scholarly journals Proof of Fermat's Last Theorem By Infinite Descent

2020 ◽  
Author(s):  
Djamel Himane
Keyword(s):  
20Th Century ◽  
Elementary Function ◽  
Mathematical Problem ◽  
History Of Mathematics ◽  
Peano Arithmetic ◽  
Basic Principles ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
History Of ◽  
Infinite Descent

Fermat's last theorem, one of the most challenging theories in the history of mathematics, has been conjectured by French lawyer Pierre de Verma in 1637. Since then, it wasconsidered the most difficult and unsolvable mathematical problem. However, more than three centuries later, a first proof was proposed by the British mathematician Andrew Wiles in 1994, relying on 20th-century techniques. Wiles's proof is based on elliptic (oval) curves that were not available at the time when the theory was first proposed. Most mathematicians argued that it was impossible to prove Fermat's theorem according to basic principles of arithmetic, though Harvey Friedman's grand conjecture states that mathematical theorems, including Fermat's Last Theorem, can be solved in very weak systems such as the Elementary Function Arithmetic (EFA). Friedman's grand conjecture states that "every theorem published in the journal, Annals of Mathematics, whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA, which is the weak fragment of Peano Arithmetic based on the usual quantifier free axioms for 0,1,+,x, exp, together with thescheme of induction for all formulas in the language all of whose quantifiers are bounded." *

scholarly journals Proving the Goldbach’s conjecture

2019 ◽  
Author(s):  
Ninh Khac Son
Keyword(s):  
Number Theory ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

Goldbach’s conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states:”Every even integer greater than 2 can be expressed as the sum of two primes”.Manuscript content: Prove that Goldbach’s conjecture is correct.

scholarly journals Chen Jingrun, China's famous mathematician: devastated by brain injuries on the doorstep to solving a fundamental mathematical puzzle

2016 ◽  
Vol 41 (1) ◽  
pp. E11
Author(s):  
Ting Lei ◽  
Evgenii Belykh ◽  
Alexander B. Dru ◽  
Kaan Yagmurlu ◽  
Ali M. Elhadi ◽  
...  
Keyword(s):  
Number Theory ◽  
Cultural Revolution ◽  
Brain Injuries ◽  
Political Repression ◽  
Brain Concussion ◽  
Chinese Youth ◽  
Chen’S Theorem ◽  
Goldbach’S Conjecture ◽  
Final Work ◽  
Goldbach's Conjecture

Chen Jingrun (1933–1996), perhaps the most prodigious mathematician of his time, focused on the field of analytical number theory. His work on Waring's problem, Legendre's conjecture, and Goldbach's conjecture led to progress in analytical number theory in the form of “Chen's Theorem,” which he published in 1966 and 1973. His early life was ravaged by the Second Sino-Japanese War and the Chinese Cultural Revolution. On the verge of solving Goldbach's conjecture in 1984, Chen was struck by a bicyclist while also bicycling and suffered severe brain trauma. During his hospitalization, he was also found to have Parkinson's disease. Chen suffered another serious brain concussion after a fall only a few months after recovering from the bicycle crash. With significant deficits, he remained hospitalized for several years without making progress while receiving modern Western medical therapies. In 1988 traditional Chinese medicine experts were called in to assist with his treatment. After a year of acupuncture and oxygen therapy, Chen could control his basic bowel and bladder functions, he could walk slowly, and his swallowing and speech improved. When Chen was unable to produce complex work or finish his final work on Goldbach's conjecture, his mathematical pursuits were taken up vigorously by his dedicated students. He was able to publish Youth Math, a mathematics book that became an inspiration in Chinese education. Although he died in 1996 at the age of 63 after surviving brutal political repression, being deprived of neurological function at the very peak of his genius, and having to be supported by his wife, Chen ironically became a symbol of dedication, perseverance, and motivation to his students and associates, to Chinese youth, to a nation, and to mathematicians and scientists worldwide.

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.

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