When is na member of a Pythagorean Triple?
... where x and y are close, co-prime and of opposite parity. On the other hand a number can be the hypotenuse of an integer right-angled triangle iff all of its prime factors are of the form 4k + 1. Notation A triple of positive integers {x, y, z} is said to be a Pythagorean Triple (PT) if it satisfies ...
... where x and y are close, co-prime and of opposite parity. On the other hand a number can be the hypotenuse of an integer right-angled triangle iff all of its prime factors are of the form 4k + 1. Notation A triple of positive integers {x, y, z} is said to be a Pythagorean Triple (PT) if it satisfies ...
Congruence Properties of the Function that Counts Compositions
... Encyclopedia [8]; one can find numerous references there. Congruence properties of b(n) modulo powers of 2 were first observed by R. F. Churchhouse [5] (the main congruence was given without a proof as a conjecture). This conjecture was later proved by H. Gupta [6] and independently by Ø. Rødseth [7 ...
... Encyclopedia [8]; one can find numerous references there. Congruence properties of b(n) modulo powers of 2 were first observed by R. F. Churchhouse [5] (the main congruence was given without a proof as a conjecture). This conjecture was later proved by H. Gupta [6] and independently by Ø. Rødseth [7 ...
Countability - Computer Science
... • Need to handle negative rationals (as in showing Int countable). ...
... • Need to handle negative rationals (as in showing Int countable). ...
PERFECT NUMBERS - Dartmouth Math Home
... We should not leave unmentioned the principal numbers . . . those which are called “perfect numbers.” These have parts which are neither larger nor smaller than the number itself, such as the number six, whose parts, three, two, and one, add up to exactly the same sum as the number itself. For the s ...
... We should not leave unmentioned the principal numbers . . . those which are called “perfect numbers.” These have parts which are neither larger nor smaller than the number itself, such as the number six, whose parts, three, two, and one, add up to exactly the same sum as the number itself. For the s ...
![[Part 1]](http://s1.studyres.com/store/data/008795849_1-075cc9ba198c265a8a5f7353865fded1-300x300.png)
On Sets of Premises - Matematički Institut SANU
... guage of propositional logic or a first-order language. Instead of the turnstile ⊢ Gentzen writes → (which is more commonly used nowadays for the binary connective of implication; we use it below, as usual, for separating the sources and targets of arrows in categories), for A and B he uses Gothic l ...
... guage of propositional logic or a first-order language. Instead of the turnstile ⊢ Gentzen writes → (which is more commonly used nowadays for the binary connective of implication; we use it below, as usual, for separating the sources and targets of arrows in categories), for A and B he uses Gothic l ...
A Contraction-free and Cut-free Sequent Calculus for
... From the point of view of Hilbert systems, propositional dynamic logic is well-defined. Indeed, there are several equivalent axiomatisations of P DL (see for example [4, 7]), each of which is obtained by adding to classical propositional logic: (i) the distribution axiom schema, that now has the form ...
... From the point of view of Hilbert systems, propositional dynamic logic is well-defined. Indeed, there are several equivalent axiomatisations of P DL (see for example [4, 7]), each of which is obtained by adding to classical propositional logic: (i) the distribution axiom schema, that now has the form ...
Notes on the large sieve
... In section LS1, we give Gallagher’s quick proof of the analytic result (without the best constant) as well as the proof by Montgomery and Vaughan of the optimal result. The modern arithmetic version, updating the results of Linnik and Rényi, is derived in section LS2: it is only here that the reaso ...
... In section LS1, we give Gallagher’s quick proof of the analytic result (without the best constant) as well as the proof by Montgomery and Vaughan of the optimal result. The modern arithmetic version, updating the results of Linnik and Rényi, is derived in section LS2: it is only here that the reaso ...
Proving algebraic inequalities
... Obviously, the set of solutions of the last inequality is the interval (, 3). To prove an inequality is to determine whether the inequality is always true for all the values of the variables on a certain set of numbers. Example: Prove that x( x 1) 0 , for all positive values of x. Solution: ...
... Obviously, the set of solutions of the last inequality is the interval (, 3). To prove an inequality is to determine whether the inequality is always true for all the values of the variables on a certain set of numbers. Example: Prove that x( x 1) 0 , for all positive values of x. Solution: ...
Godel incompleteness
... them interesting, and therefore popular... but also misinterpreted! First of all, the theorem does not say that every axiomatic system is necessarily incomplete: for example, absolute geometry with the following rules is incomplete: 1. Any two points can be joined by a straight line. 2. Any straight ...
... them interesting, and therefore popular... but also misinterpreted! First of all, the theorem does not say that every axiomatic system is necessarily incomplete: for example, absolute geometry with the following rules is incomplete: 1. Any two points can be joined by a straight line. 2. Any straight ...
![[hal-00574623, v2] Averaging along Uniform Random Integers](http://s1.studyres.com/store/data/019969824_1-25527940ea4f317ec31969269e4745aa-300x300.png)
[hal-00574623, v2] Averaging along Uniform Random Integers
... leading digit d ∈ {1, 2, . . . , 9} is log10 (1 + 1/d). More generally, defining the mantissa M (x) of a positive real number x as the only real number in [1, 10[ such that x = M (x)10k for some integer k, Benford’s law states that for any 1 ≤ α < β < 10, the proportion of numbers whose mantissa lie ...
... leading digit d ∈ {1, 2, . . . , 9} is log10 (1 + 1/d). More generally, defining the mantissa M (x) of a positive real number x as the only real number in [1, 10[ such that x = M (x)10k for some integer k, Benford’s law states that for any 1 ≤ α < β < 10, the proportion of numbers whose mantissa lie ...
na.
... validity problems for TCg. TCb and TCf are not members ot DTIME(c n ). where n. is the size (I) of the formula. The proof is by reduction to one letter PDL. Let us call PDL- the restriction of PDL obtained by restricting oneself to a Bingle basic program ( 1-) and not allowing sequential eomposition ...
... validity problems for TCg. TCb and TCf are not members ot DTIME(c n ). where n. is the size (I) of the formula. The proof is by reduction to one letter PDL. Let us call PDL- the restriction of PDL obtained by restricting oneself to a Bingle basic program ( 1-) and not allowing sequential eomposition ...
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.