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An Ancient Diophantine Equation with applications to Numerical
... k = 3, Eq. (2) dates back to Diophantus who gave a numerical solution in his Arithmetica [2, p. 173]. The complete solution of (2) with k = 3 is readily obtained and is already known. When k = 4, a single numerical solution of Eq. (2) in positive rational numbers was obtained by Fermat [2, pp. 319-2 ...
... k = 3, Eq. (2) dates back to Diophantus who gave a numerical solution in his Arithmetica [2, p. 173]. The complete solution of (2) with k = 3 is readily obtained and is already known. When k = 4, a single numerical solution of Eq. (2) in positive rational numbers was obtained by Fermat [2, pp. 319-2 ...
Structure and Randomness in the prime numbers
... that every odd number greater than 1 can be written as the sum of five primes or less, by modifying Vinogradov’s argument. In 1742, Christian Goldbach conjectured that in fact every odd number n greater than 5 should be the sum of three primes. This is currently only known for n larger than 101346 ( ...
... that every odd number greater than 1 can be written as the sum of five primes or less, by modifying Vinogradov’s argument. In 1742, Christian Goldbach conjectured that in fact every odd number n greater than 5 should be the sum of three primes. This is currently only known for n larger than 101346 ( ...
pdf - at www.arxiv.org.
... Let note that the right terms of several congruencies could be considered as negative integers, in this case the result of the lemma is still valid (that is, supposing bi > 0, we would have: d ≡ −1 · bi (mod pi ) for some i ∈ [1..k], and the correspondent terms of the expession of the solution would ...
... Let note that the right terms of several congruencies could be considered as negative integers, in this case the result of the lemma is still valid (that is, supposing bi > 0, we would have: d ≡ −1 · bi (mod pi ) for some i ∈ [1..k], and the correspondent terms of the expession of the solution would ...
Lecture Notes on Primality Testing
... only with a = 2 is sometimes used to produce “industrial grade” primes. This simplified version makes only 22 errors in the first 10,000 integers. It has also been proved for this version that lim Pr[Error on random b-bit number] → 0. ...
... only with a = 2 is sometimes used to produce “industrial grade” primes. This simplified version makes only 22 errors in the first 10,000 integers. It has also been proved for this version that lim Pr[Error on random b-bit number] → 0. ...
ON THE ERD¨OS-STRAUS CONJECTURE
... decomposition with the smallest a possible is exhibited in the equality ...
... decomposition with the smallest a possible is exhibited in the equality ...
Maximizing the number of nonnegative subsets, SIAM J. Discrete
... all the proper subsets containing x1 , together with the empty set, have a nonnegative sum. It is also not hard to see that this is best possible, since for every subset A either A or its complement {x1 , . . . , xn }\A must have a negative sum. Now a new question arises: Suppose it is known that ev ...
... all the proper subsets containing x1 , together with the empty set, have a nonnegative sum. It is also not hard to see that this is best possible, since for every subset A either A or its complement {x1 , . . . , xn }\A must have a negative sum. Now a new question arises: Suppose it is known that ev ...
DIOPHANTINE APPROXIMATION OF COMPLEX NUMBERS
... has infinitely many non-trivial solutions x, y ∈ Z[i D]. 2. Geometry of Numbers Hilde Gintner [5] mainly refers to two theorems of Minkowksi [15] coming from his famous theory of ’Geometry of Numbers’. Those theorems can also be found in [2], where J.W.S. Cassels so to say translated them to a more ...
... has infinitely many non-trivial solutions x, y ∈ Z[i D]. 2. Geometry of Numbers Hilde Gintner [5] mainly refers to two theorems of Minkowksi [15] coming from his famous theory of ’Geometry of Numbers’. Those theorems can also be found in [2], where J.W.S. Cassels so to say translated them to a more ...
Some sufficient conditions of a given series with rational terms
... a given real number is an irrational number or a transcendental number such as well known Roth theorem but seems to be lack of practical test just as various convenient test in series theory. The purpose of this paper is to propose some sufficient conditions for convenient use in determining if a gi ...
... a given real number is an irrational number or a transcendental number such as well known Roth theorem but seems to be lack of practical test just as various convenient test in series theory. The purpose of this paper is to propose some sufficient conditions for convenient use in determining if a gi ...
FP3: Complex Numbers - Schoolworkout.co.uk
... This result applies for any positive or negative power n. Note: We will be able to prove this result more formally next lesson. Example: z 3 i . a) Find z 2 and z3 . b) Find z8 and z 3 Solution: First write z in modulus-argument form: ...
... This result applies for any positive or negative power n. Note: We will be able to prove this result more formally next lesson. Example: z 3 i . a) Find z 2 and z3 . b) Find z8 and z 3 Solution: First write z in modulus-argument form: ...
ON THE FRACTIONAL PARTS OF LACUNARY SEQUENCES
... Proof of Theorem 5. Fix u in {0, 1, .√. . , 9} and set tn = 10n/2 , ν = −u/10 in Theorem 1. Note that 0.4625 > 1/( 10 − 1). It follows from Theorem 1 that there is a positive ξ such that {ξ 10n/2 −√u/10} < 0.4625 for each integer n ≥ 0. Hence {ξ 10k − u/10} < 1/2 and {(ξ 10)10k − u/10} < 1/2 for eac ...
... Proof of Theorem 5. Fix u in {0, 1, .√. . , 9} and set tn = 10n/2 , ν = −u/10 in Theorem 1. Note that 0.4625 > 1/( 10 − 1). It follows from Theorem 1 that there is a positive ξ such that {ξ 10n/2 −√u/10} < 0.4625 for each integer n ≥ 0. Hence {ξ 10k − u/10} < 1/2 and {(ξ 10)10k − u/10} < 1/2 for eac ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".