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Clement's congruence-based theorem characterizes the twin primes pairs of the form (, +) through the following conditions: [()! +] ((+)), +P. A. Clement's original 1949 paper [2] provides a proof of this interesting elementary number theoretic criteria for twin primality based on Wilson's theorem.
An equivalent formulation in this context is the following: [4] A congruence relation on an algebra A is a subset of the direct product A × A that is both an equivalence relation on A and a subalgebra of A × A. The kernel of a homomorphism is always a congruence. Indeed, every congruence arises as a kernel.
Each of the Archimedes' quadruplets (green) have equal area to each other and to Archimedes' twin circles. In geometry, Archimedes' quadruplets are four congruent circles associated with an arbelos. Introduced by Frank Power in the summer of 1998, each have the same area as Archimedes' twin circles, making them Archimedean circles. [1] [2] [3]
The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest. The relation " ∼ {\displaystyle \sim } is finer than ≈ {\displaystyle \approx } " on the collection of all equivalence relations on a fixed set is itself a partial order ...
Removing five axioms mentioning "plane" in an essential way, namely I.4–8, and modifying III.4 and IV.1 to omit mention of planes, yields an axiomatization of Euclidean plane geometry. Hilbert's axioms, unlike Tarski's axioms, do not constitute a first-order theory because the axioms V.1–2 cannot be expressed in first-order logic.
Similarly, the ascending chain condition is equivalent to P being converse well-founded (again, assuming dependent choice): every nonempty subset of P has a maximal element (the maximal condition or maximum condition).
The six numbered statements also repeat some of what is said in the paragraphs. My recommendation, remove the six "lines" and organize the 4 short paragraphs to cover the conditions for congruency based on additional information (5 short paragraphs when you add back in the HL postulate). JackOL31 04:13, 2 December 2009 (UTC)
In mathematics, Gauss congruence is a property held by certain sequences of integers, including the Lucas numbers and the divisor sum sequence. Sequences satisfying this property are also known as Dold sequences, Fermat sequences, Newton sequences, and realizable sequences. [1]