n - Stanford University
... Theorem: If ¬Q → ¬P, then P → Q. Proof: By contradiction. Assume that ¬Q → ¬P, but that P → Q is false. Since P → Q is false, it must be true that P is true and ¬Q is true. Since ¬Q is true and ¬Q → ¬P, we know that ¬P is true. But this means that we have shown P and ¬P, which is impossible. We have ...
... Theorem: If ¬Q → ¬P, then P → Q. Proof: By contradiction. Assume that ¬Q → ¬P, but that P → Q is false. Since P → Q is false, it must be true that P is true and ¬Q is true. Since ¬Q is true and ¬Q → ¬P, we know that ¬P is true. But this means that we have shown P and ¬P, which is impossible. We have ...
Two statements that are equivalent to a
... there always exists a prime number p such that n < p < 2n − 2 (another formulation of this theorem is that for every n > 1 there always exists a prime number p such that n < p < 2n). This statement, which had been conjectured by Joseph Bertrand in 1845, was first proved by P. L. Chebyshev in 1850. A ...
... there always exists a prime number p such that n < p < 2n − 2 (another formulation of this theorem is that for every n > 1 there always exists a prime number p such that n < p < 2n). This statement, which had been conjectured by Joseph Bertrand in 1845, was first proved by P. L. Chebyshev in 1850. A ...
the prime number theorem for rankin-selberg l
... A remarkable feature of this corollary is that it describes the orthogonality of a π (n) and aπ0 (n) in three cases with different main terms. It is thus in a more precise form than Selberg’s Conjecture 1.2. Moreover, one can see from the last case of Corollary 3.3 that the Dirichlet series on the r ...
... A remarkable feature of this corollary is that it describes the orthogonality of a π (n) and aπ0 (n) in three cases with different main terms. It is thus in a more precise form than Selberg’s Conjecture 1.2. Moreover, one can see from the last case of Corollary 3.3 that the Dirichlet series on the r ...
sets of uniqueness and sets of multiplicity
... Let 0= l/£. If y(u) does not tend to zero for u= <*>,there exists a number X (l/0=Xgl) such that the series 52" sin2 xX0" converges. Once this result is proved, Theorem II is a direct consequence of the theory of Pisot. Since, however, this theory is unnecessarily general for our particular purpose, ...
... Let 0= l/£. If y(u) does not tend to zero for u= <*>,there exists a number X (l/0=Xgl) such that the series 52" sin2 xX0" converges. Once this result is proved, Theorem II is a direct consequence of the theory of Pisot. Since, however, this theory is unnecessarily general for our particular purpose, ...
PEN A9 A37 O51
... wonderful to find such pearls of number theory available online as public domain- to paraphrase Newton, it really helps us keep our balance on the giants’ shoulders). Still, Pillai’s methods shown in the article mentioned above aren’t efficient in this case. Still, in that time it was already prove ...
... wonderful to find such pearls of number theory available online as public domain- to paraphrase Newton, it really helps us keep our balance on the giants’ shoulders). Still, Pillai’s methods shown in the article mentioned above aren’t efficient in this case. Still, in that time it was already prove ...
MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS (III)
... (c) (Lagrange’s theorem) Every n ∈ N is a sum of four squares of integers. Those integers Tx = x(x + 1)/2 with x ∈ Z are called triangular numbers. Note that Tx = T−x−1 and 8Tx + 1 = (2x + 1)2 . In 1638 P. Fermat asserted that each n ∈ N can be written as a sum of three triangular numbers (equivalen ...
... (c) (Lagrange’s theorem) Every n ∈ N is a sum of four squares of integers. Those integers Tx = x(x + 1)/2 with x ∈ Z are called triangular numbers. Note that Tx = T−x−1 and 8Tx + 1 = (2x + 1)2 . In 1638 P. Fermat asserted that each n ∈ N can be written as a sum of three triangular numbers (equivalen ...
Feedback, Control, and the Distribution of Prime Numbers
... twice previously. The model seems to have been forgotten by the number theory community, but the distribution of primes is mentioned as an application in the differential equation literature. (See, for example, [7, p. 237] or [16].) We endeavor to contribute a deeper understanding of the connection ...
... twice previously. The model seems to have been forgotten by the number theory community, but the distribution of primes is mentioned as an application in the differential equation literature. (See, for example, [7, p. 237] or [16].) We endeavor to contribute a deeper understanding of the connection ...
Chapter 4
... Definition. Suppose that a, b ∈ Z and a 6= 0. Then we say that a divides b, denoted by a | b, if there exists c ∈ Z such that b = ac. In this case, we also say that a is a divisor of b, or b is a multiple of a. Example 4.1.1. For every a ∈ Z \ {0}, a | a and a | −a. Example 4.1.2. For every a ∈ Z, 1 ...
... Definition. Suppose that a, b ∈ Z and a 6= 0. Then we say that a divides b, denoted by a | b, if there exists c ∈ Z such that b = ac. In this case, we also say that a is a divisor of b, or b is a multiple of a. Example 4.1.1. For every a ∈ Z \ {0}, a | a and a | −a. Example 4.1.2. For every a ∈ Z, 1 ...
ARITHMETIC TRANSLATIONS OF AXIOM SYSTEMS
... Con(N) as an axiom) and that therefore the relative consistency of N' to N# can be proved in number theory. It is also observed that if N is co-consistent, then iV# is consistent. The same method can be applied, for similarly related systems S and S', to prove the relative consistency of S' to S#, a ...
... Con(N) as an axiom) and that therefore the relative consistency of N' to N# can be proved in number theory. It is also observed that if N is co-consistent, then iV# is consistent. The same method can be applied, for similarly related systems S and S', to prove the relative consistency of S' to S#, a ...
Tiling Proofs of Recent Sum Identities Involving Pell Numbers
... will consider the sequence of numbers denoted pn and defined as pn = Pn+1 for n ≥ −1. The motivation for considering pn is simple: pn counts the number of tilings of a board of length n using white squares, black squares, and gray dominoes. Thus, for example, p0 = 1 counts the empty tiling, p1 = 2 b ...
... will consider the sequence of numbers denoted pn and defined as pn = Pn+1 for n ≥ −1. The motivation for considering pn is simple: pn counts the number of tilings of a board of length n using white squares, black squares, and gray dominoes. Thus, for example, p0 = 1 counts the empty tiling, p1 = 2 b ...
FACTORING WITH CONTINUED FRACTIONS, THE PELL
... possible. Once the number of completely factored integers exceeds the size of the factor base, we can nd a product of them which is a perfect square. With a little luck this yields a non-trivial factor of our given number (by the observations from the introduction). The crucial property of the valu ...
... possible. Once the number of completely factored integers exceeds the size of the factor base, we can nd a product of them which is a perfect square. With a little luck this yields a non-trivial factor of our given number (by the observations from the introduction). The crucial property of the valu ...
Proofs
... Thus, n is even because it is divisible by 2. It follows that the integer n also must be even. So both m and n are even and therefore both are divisible by 2. But this fact contradicts the assumption that we have chosen m and n to have no common divisors. This contradiction leads us to conclude that ...
... Thus, n is even because it is divisible by 2. It follows that the integer n also must be even. So both m and n are even and therefore both are divisible by 2. But this fact contradicts the assumption that we have chosen m and n to have no common divisors. This contradiction leads us to conclude that ...
On nonexistence of an integer regular polygon∗
... The problem of existence of an integer regular polygon is equivalent to the following: does there exist an integer regular polygon whose coordinates are natural, i.e. rational numbers. It is very important to note the following: if we consider a regular polygon with n sides, whereby n is not a prime ...
... The problem of existence of an integer regular polygon is equivalent to the following: does there exist an integer regular polygon whose coordinates are natural, i.e. rational numbers. It is very important to note the following: if we consider a regular polygon with n sides, whereby n is not a prime ...
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"".