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There exists a positive integer such that every sufficiently large integer is the sum of at most primes. It follows that there exists a positive integer such that every integer is a sum of at most primes. The smallest proven value of is known as the Schnirelmann constant.Schnirelmann's theorem can be proved using Mann's theorem,although Schnirelmann used the weaker inequalitywhere , , and is the Schnirelmann density. Let be the set of primes, together with 0 and 1, and let . Using a sophisticated version of the inclusion-exclusion principle, Schnirelmann showed that although , . By repeated applications of Mann's theorem, the sum of copies of satisfies . Thus, if , the sum of copies of has Schnirelmann density 1, and so contains all positive integers.

Mann's theorem is a theorem widely circulated as the " conjecture" that was subsequently proven by Mann (1942). It states that if and are sets of integers each containing 0, thenHere, denotes the direct sum, i.e., , and is the Schnirelmann density.Mann's theorem is optimal in the sense that satisfies .Mann's theorem implies Schnirelmann's theorem as follows. Let , then Mann's theorem proves that , so as more and more copies of the primes are included, the Schnirelmann density increases at least linearly, and so reaches 1 with at most copies of the primes. Since the only sets with Schnirelmann density 1 are the sets containing all positive integers, Schnirelmann's theorem follows.

van der Waerden's theorem is a theorem about the existence of arithmetic progressions in sets. The theorem can be stated in four equivalent forms. 1. If , then some contains arbitrarily long arithmetic progressions (Baudet's conjecture). 2. For all positive integers and , there exists a constant such that if and , then some set contains an arithmetic progression of length . 3. If is an infinite sequence of integers satisfying for some , then the sequence contains arbitrarily long arithmetic progressions. 4. For all positive integers and , there is a constant such that if and , , ..., satisfies , then of the numbers , , ..., are in arithmetic progression. The constants are called van der Waerden numbers, and no formula for is known. van der Waerden's theorem is a corollary of Szemerédi's theorem...

One form of van der Waerden's theorem states that for all positive integers and , there exists a constant such that if and , then some set contains an arithmetic progression of length . The least possible value of is known as a van der Waerden number. The only nontrivial van der Waerden numbers that are known exactly are summarized in the following table. As shown in the table, the first few values of for , 2, ... are 1, 3, 9, 35, 178, 1132, ... (OEIS A005346), the last of which is due to M. Kouril and J. L. Paul in 2007 (Kouril and Paul 2008).345629351781132327476Shelah (1988) proved that van der Waerden's numbers are primitiverecursive. It is known that(1)and that(2)for some constants and . In 1998, T. Gowers announced that he has proved the general result(3)(Gowers 2001). Berlekamp (1968) showed that for a prime,(4)A probabilistic argument using the Lovászlocal lemma shows that(5)New lower bounds have been given..

An unsolved problem in mathematics attributed to Lehmer (1933) that concerns the minimum Mahler measure for a univariate polynomial that is not a product of cyclotomic polynomials. Lehmer (1933) conjectured that if is such an integer polynomial, then(1)(2)where , denoted by Lehmer (1933) and by Hironaka (2009), is the largest positive root of this polynomial. The roots of this polynomial, plotted in the left figure above, are very special, since 8 of the 10 lie on the unit circle in the complex plane. The roots of the polynomials (represented by half their coefficients) giving the two next smallest known Mahler measures are also illustrated above (Mossinghoff 1998, p. S11).The best current bound is that of Smyth (1971), who showed that , where is a nonzero nonreciprocal polynomial that is not a product of cyclotomic polynomials (Everest 1999), and is the real root of . Generalizations of Smyth's result have been constructed by Lloyd-Smith..

A Thue equation is a Diophantine equation of the formin terms of an irreducible polynomial of degree having coefficients for which solutions in integers and are sought for each given constant with .Thue (1909) proved that such an equation has only finitely many solutions, but it was not until much later that Tzanakis and de Weger (1989) gave a practical algorithm for finding bounds on and . Although these bounds can be astronomically large in some cases, they are typically small enough to allow an exhaustive search for all solutions.

For a polynomial , the Mahler measure of is defined by(1)Using Jensen's formula, it can be shown that for ,(2)(Borwein and Erdélyi 1995, p. 271).Specific cases are given by(3)(4)(5)(Borwein and Erdélyi 1995, p. 272).A product of cyclotomic polynomials has Mahler measure 1. The Mahler measure of an integer polynomial in variables gives the topological entropy of a -dynamical system canonically associated to the polynomial.Lehmer's Mahler measure problem conjectures that a particular univariate polynomial has the smallest possible Mahler measure other than 1.

Szemerédi's theorem states that every sequence of integers that has positive upper Banach density contains arbitrarily long arithmetic progressions.A corollary states that, for any positive integer and positive real number , there exists a threshold number such that for every subset of with cardinal number larger than contains a -term arithmetic progression. van der Waerden's Theorem follows immediately by setting . The best bounds for van der Waerden numbers are derived from bounds for in Szemerédi's theorem.Szemerédi's theorem was conjectured by Erdős and Turán (1936). Roth (1953) proved the case , and was mentioned in his Fields Medal citation. Szemerédi (1969) proved the case , and the general theorem in 1975 as a consequence of Szemerédi's regularity lemma (Szemerédi 1975a), for which he collected a $1000 prize from Erdos. Fürstenberg and Katznelson (1979) proved..

The Banach density of a set of integers is defined asif the limit exists. If the is replaced with or , then the result is known as the upper or lower Banach density, respectively.In the ergodic theory approach to Szemerédi's theorem, Banach density must be used. (Although the statements of Szemerédi's theorem with different types of density are equivalent, the proofs are not easily converted from one density type to the other.)

A basis for the real numbers , considered as a vector space over the rationals , i.e., a set of real numbers such that every real number has a unique representation of the formwhere is rational and depends on .The axiom of choice is equivalent to the statement: "Every vector space has a vector space basis," and this is the only justification for the existence of a Hamel basis.

The ordered pair , where is the number of real embeddings of the number field and is the number of complex-conjugate pairs of embeddings. The degree of the number field is .

The Bombieri -norm of a polynomial(1)is defined by(2)where is a binomial coefficient. The most remarkable feature of Bombieri's norm is that given polynomials and such that , then Bombieri's inequality(3)holds, where is the degree of , and is the degree of either or . This theorem captures the heuristic that if and have big coefficients, then so does , i.e., there can't be too much cancellation.

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