Multiplicities and Enumeration of Semidualizing Modules
... Christensen, Striuli, and Veliche [5]. Also, note that there exist rings of codimension greater than 2 such that S0 (R) = {[R], [D]}; see, e.g. Lemma 3.1. Our proof of Theorem 1.3 hinges on the computation of lenRP (CP ) where [C] ∈ S0 (R) and P ∈ Ass(R), which we do in Lemma 5.1. In a sense, this i ...
... Christensen, Striuli, and Veliche [5]. Also, note that there exist rings of codimension greater than 2 such that S0 (R) = {[R], [D]}; see, e.g. Lemma 3.1. Our proof of Theorem 1.3 hinges on the computation of lenRP (CP ) where [C] ∈ S0 (R) and P ∈ Ass(R), which we do in Lemma 5.1. In a sense, this i ...
2.9.2 Problems P10 Try small prime numbers first. p p2 + 2 2 6 3 11
... The condition in the problem is the same as saying that every prime number p divides m. This can happen only if m = 0, that is, only if n = 1. Hence n = 1 is the only positive integer with the given property. P15 We try to find a pattern. ...
... The condition in the problem is the same as saying that every prime number p divides m. This can happen only if m = 0, that is, only if n = 1. Hence n = 1 is the only positive integer with the given property. P15 We try to find a pattern. ...
For screen
... n 6≡ 0 (mod 3) and n 6≡ 0 (mod 4). Furthermore, if such solution exists, known results on Catalan’s equation imply that n must be odd. Indeed, if q0 n = 2m, then xm 0 − y1 = −1 for some positive integer y1 dividing y, and this is impossible since x0 < 1011 , by a result of Hyyrö (see [13], pages 26 ...
... n 6≡ 0 (mod 3) and n 6≡ 0 (mod 4). Furthermore, if such solution exists, known results on Catalan’s equation imply that n must be odd. Indeed, if q0 n = 2m, then xm 0 − y1 = −1 for some positive integer y1 dividing y, and this is impossible since x0 < 1011 , by a result of Hyyrö (see [13], pages 26 ...
The equational theory of N, 0, 1, +, ×, ↑ is decidable, but not finitely
... school are complete for showing all arithmetic equations valid for the natural numbers. The answer to this question has occupied many prestigious mathematicians over half a century, that gave the answer for various subsystems, the most intriguing one being the one involving a constant for the number ...
... school are complete for showing all arithmetic equations valid for the natural numbers. The answer to this question has occupied many prestigious mathematicians over half a century, that gave the answer for various subsystems, the most intriguing one being the one involving a constant for the number ...
ON CONGRUENT NUMBERS WITH THREE PRIME FACTORS
... will assume prime values infinitely often. In order to obtain q3 , r3 prime numbers from these two forms, we must have a odd. By Lemma 2 the number n = p3 q3 r3 will be congruent. All of the examples of congruent numbers mentioned in the introduction have p3 = 3, but we can generate examples for any ...
... will assume prime values infinitely often. In order to obtain q3 , r3 prime numbers from these two forms, we must have a odd. By Lemma 2 the number n = p3 q3 r3 will be congruent. All of the examples of congruent numbers mentioned in the introduction have p3 = 3, but we can generate examples for any ...
irrationality and transcendence 4. continued fractions.
... (in a sense to be explained later) to a given real number by the use of continued fractions. We shall start with the Euclidean algorithm for computing the greatest common divisor of two (positive) integers. For example, beginning with 95 and 37 we have ...
... (in a sense to be explained later) to a given real number by the use of continued fractions. We shall start with the Euclidean algorithm for computing the greatest common divisor of two (positive) integers. For example, beginning with 95 and 37 we have ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.