The Abundancy Index of Divisors of Odd Perfect Numbers
... is an integer (because gcd(qi αi , σ(qi αi )) = 1). Suppose ρi = 1. Then σ(N/qi αi ) = qi αi and σ(qi αi ) = 2N/qi αi . Since N is an odd perfect number, qi is odd, whereupon we have an odd αi by considering parity conditions from the last equation. But this means that qi is the Euler prime q, and w ...
... is an integer (because gcd(qi αi , σ(qi αi )) = 1). Suppose ρi = 1. Then σ(N/qi αi ) = qi αi and σ(qi αi ) = 2N/qi αi . Since N is an odd perfect number, qi is odd, whereupon we have an odd αi by considering parity conditions from the last equation. But this means that qi is the Euler prime q, and w ...
(Convenient) Numbers - UGA Math Department
... We can similarly prove statements of type (1) for all 65 numbers in Table 1. In this article’s last section, we show that the Geometry of Numbers can also be used to prove similar statements for some non-convenient numbers. For example, n = 14 is not a convenient number as one can show (by elementar ...
... We can similarly prove statements of type (1) for all 65 numbers in Table 1. In this article’s last section, we show that the Geometry of Numbers can also be used to prove similar statements for some non-convenient numbers. For example, n = 14 is not a convenient number as one can show (by elementar ...
THE p–ADIC ORDER OF POWER SUMS, THE ERD
... Let Vp (m) = d. Then, as in Theorem 3, we may write m = qpd + a0 k=0 0 ≤ q 6≡ a0 (mod p). Remark 7. If m ≡ −1 (mod p), then the equalities Vp (m) = Vp (m + 1) = vp (m + 1) hold. Indeed, write m in base p as m = . . . ah (p − 1)(p − 1) . . . (p − 1)p , with ah 6= p − 1 so that Vp (m) = h. Notice that ...
... Let Vp (m) = d. Then, as in Theorem 3, we may write m = qpd + a0 k=0 0 ≤ q 6≡ a0 (mod p). Remark 7. If m ≡ −1 (mod p), then the equalities Vp (m) = Vp (m + 1) = vp (m + 1) hold. Indeed, write m in base p as m = . . . ah (p − 1)(p − 1) . . . (p − 1)p , with ah 6= p − 1 so that Vp (m) = h. Notice that ...
29(2)
... if n and m are of the same parity, then expansion (2.11) will only involve Bernoulli polynomials of even index. If n and m are of opposite parity, then expansion (2.11) will only involve Bernoulli polynomials of odd index. If we define ...
... if n and m are of the same parity, then expansion (2.11) will only involve Bernoulli polynomials of even index. If n and m are of opposite parity, then expansion (2.11) will only involve Bernoulli polynomials of odd index. If we define ...
THE LOGIC OF QUANTIFIED STATEMENTS
... • add quantifiers, words that refer to quantities such as “some” or “all” and tell for how many elements a given predicate is true. • e.g., For some integer x, x is divisible by 5 • e.g., For all integer x, x is divisible by 5 • e.g., there exists two integer x, such that x is divisible by 5. • All ...
... • add quantifiers, words that refer to quantities such as “some” or “all” and tell for how many elements a given predicate is true. • e.g., For some integer x, x is divisible by 5 • e.g., For all integer x, x is divisible by 5 • e.g., there exists two integer x, such that x is divisible by 5. • All ...
Shape is a Non-Quantifiable Physical Dimension
... though the curving-in by rotation can be turned into the curving-out (and the angleinward into the angle-outward), and they are in this sense identical shapes, they are nonetheless as shape segments different. If a curving-in is via one of its end points joined to the end point of another curving-in ...
... though the curving-in by rotation can be turned into the curving-out (and the angleinward into the angle-outward), and they are in this sense identical shapes, they are nonetheless as shape segments different. If a curving-in is via one of its end points joined to the end point of another curving-in ...
THE ULTRAPRODUCT CONSTRUCTION 1. Introduction The
... The ultraproduct construction is a uniform method of building models of first order theories which has applications in many areas of mathematics. It is attractive because it is algebraic in nature, but preserves all properties expressible in first order logic. The idea goes back to the construction of ...
... The ultraproduct construction is a uniform method of building models of first order theories which has applications in many areas of mathematics. It is attractive because it is algebraic in nature, but preserves all properties expressible in first order logic. The idea goes back to the construction of ...
LOGICAL CONSEQUENCE AS TRUTH-PRESERVATION STEPHEN READ Abstract
... What these proofs show is that, if we suppose that 3 represents implication (real implication), then an impossible proposition implies any proposition, and if ⊃ represents implication, even a false proposition implies any other. Neither proof shows that these are sound principles about implication. ...
... What these proofs show is that, if we suppose that 3 represents implication (real implication), then an impossible proposition implies any proposition, and if ⊃ represents implication, even a false proposition implies any other. Neither proof shows that these are sound principles about implication. ...
Products of random variables and the first digit phenomenon
... Theorem 3.2 below is a direct consequence of Proposition 3.1 which is a slight generalization of a (surprisingly little known) result due to Davenport, Erdős and Le Veque [7]. Proposition 3.1 gives a general condition (involving Lp -norm) ensuring that the arithmetic mean of bounded random variables ...
... Theorem 3.2 below is a direct consequence of Proposition 3.1 which is a slight generalization of a (surprisingly little known) result due to Davenport, Erdős and Le Veque [7]. Proposition 3.1 gives a general condition (involving Lp -norm) ensuring that the arithmetic mean of bounded random variables ...
Fermat`s little theorem, Chinese Remainder Theorem
... If a and b are relatively prime if and only if there are integers m and n so that am + bn = 1. Working modulo b, this says Lemma 2. a and b are relatively prime if and only if there is an integer m such that am ≡ 1 ...
... If a and b are relatively prime if and only if there are integers m and n so that am + bn = 1. Working modulo b, this says Lemma 2. a and b are relatively prime if and only if there is an integer m such that am ≡ 1 ...
Bridge to Abstract Mathematics: Mathematical Proof and
... countably infinite collections of sets. The main emphasis here is on standard approaches to proving set inclusion (e.g., the "choose" method) and set equality (e.g., mutual inclusion), but we manage also, through the many solved examples, to anticipate additional techniques of proof that are studied ...
... countably infinite collections of sets. The main emphasis here is on standard approaches to proving set inclusion (e.g., the "choose" method) and set equality (e.g., mutual inclusion), but we manage also, through the many solved examples, to anticipate additional techniques of proof that are studied ...
Limits and Infinite Series Lecture Notes for Math 226 by´Arpád Bényi
... Infinity is a concept that we find convenient to use when describing the behavior of a function whose input or output values become arbitrarily large. We will distinguish between positive infinity (+∞) and negative infinity (−∞): +∞ or simply ∞ has to do with a quantity that becomes arbitrarily large an ...
... Infinity is a concept that we find convenient to use when describing the behavior of a function whose input or output values become arbitrarily large. We will distinguish between positive infinity (+∞) and negative infinity (−∞): +∞ or simply ∞ has to do with a quantity that becomes arbitrarily large an ...
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.