Integers, Modular Arithmetic and Cryptography
... n = 43 x 59 = 2537 (i.e. p = 43, q = 59). Everybody knows n. but nobody knows p or q – they are secret. (p-1)(q-1) = 42 x 58 = 2436 Encryption key e = 13 (must be relatively prime with 2436) (secret). Decryption key d = 937 (is the inverse of e mod (p-1)(q-1)) (public knowledge) ...
... n = 43 x 59 = 2537 (i.e. p = 43, q = 59). Everybody knows n. but nobody knows p or q – they are secret. (p-1)(q-1) = 42 x 58 = 2436 Encryption key e = 13 (must be relatively prime with 2436) (secret). Decryption key d = 937 (is the inverse of e mod (p-1)(q-1)) (public knowledge) ...
Sample pages 2 PDF
... 4 Derrick Henry Lehmer (1905–1991), son of D.N. Lehmer, student of J. Tamarkin, professor at Berkeley. See [729]. ...
... 4 Derrick Henry Lehmer (1905–1991), son of D.N. Lehmer, student of J. Tamarkin, professor at Berkeley. See [729]. ...
Full text
... and JC + 1 are a conjugate pair of factors of f„(x) and fn(x +1). Using Lemma 4 of [1] and Induction, it is not hard to prove that x + 1 Is a factor of fk if and only If 31 Ar and this property also happens to be a special case of Proposition 5(b) of [1]. Hence, we have that 6|«, which implies that ...
... and JC + 1 are a conjugate pair of factors of f„(x) and fn(x +1). Using Lemma 4 of [1] and Induction, it is not hard to prove that x + 1 Is a factor of fk if and only If 31 Ar and this property also happens to be a special case of Proposition 5(b) of [1]. Hence, we have that 6|«, which implies that ...
The Fundamentals: Algorithms, the Integers, and Matrices
... composite, it is a prime factor of q. This contradicts the assumption that p1, p2, ….., pn are all the primes. Consequently, there are infinitely many primes. This proof was given by Euclid The Elements. The proof is considered to be one of the most beautiful in all mathematics. It is the first pr ...
... composite, it is a prime factor of q. This contradicts the assumption that p1, p2, ….., pn are all the primes. Consequently, there are infinitely many primes. This proof was given by Euclid The Elements. The proof is considered to be one of the most beautiful in all mathematics. It is the first pr ...
1 Introduction - University of South Carolina
... Theorem 2. There are infinitely many composite natural numbers N , coprime to 10, with the property that if you insert any digit d ∈ {0, . . . , 9} anywhere in the decimal expansion of N , then the number created by this insertion is composite. Another problem that could be considered along similar ...
... Theorem 2. There are infinitely many composite natural numbers N , coprime to 10, with the property that if you insert any digit d ∈ {0, . . . , 9} anywhere in the decimal expansion of N , then the number created by this insertion is composite. Another problem that could be considered along similar ...
The 3n + 1 conjecture
... Figure 1: The values of an plotted against n, with a0 = 27 and a111 = 1. other cycle or end in the cycle (1, 4, 2). However, both brute force calculations for different starting values and heuristic calculations based on ’averages’ can never prove the conjecture. We cannot calculate the sequences fo ...
... Figure 1: The values of an plotted against n, with a0 = 27 and a111 = 1. other cycle or end in the cycle (1, 4, 2). However, both brute force calculations for different starting values and heuristic calculations based on ’averages’ can never prove the conjecture. We cannot calculate the sequences fo ...
The largest prime factor of a Mersenne number
... knew that if 2n − 1 is prime, then so is n prime, and that the converse does not always hold. In the 18th century, Euler showed that Euclid’s formula for perfect numbers gives rise to all even examples. (It is conjectured that there are no odd perfect numbers.) Probably because of the connection to ...
... knew that if 2n − 1 is prime, then so is n prime, and that the converse does not always hold. In the 18th century, Euler showed that Euclid’s formula for perfect numbers gives rise to all even examples. (It is conjectured that there are no odd perfect numbers.) Probably because of the connection to ...
Flat primes and thin primes
... Some interesting subclasses of primes have been identified and actively considered. These include Mersenne primes, Sophie Germain primes, Fermat primes, Cullen’s primes, Wieferich primes, primes of the form n2 + 1, of the form n! ± 1, etc. See for example [14, Chapter 5] and the references in that t ...
... Some interesting subclasses of primes have been identified and actively considered. These include Mersenne primes, Sophie Germain primes, Fermat primes, Cullen’s primes, Wieferich primes, primes of the form n2 + 1, of the form n! ± 1, etc. See for example [14, Chapter 5] and the references in that t ...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.