Every odd number greater than $1$ is the sum of at most five primes

Call a (strictly increasing) sequence R = (rn) of natural numbers regular if it satisfies the following condition: r n+1 /rn → θ ∈ R >1 ∪ {∞} and, if θ is algebraic, then R satisfies a recurrence relation whose characteristic polynomial is the minimal polynomial of θ.Our main result states that Z R = (Z, +, 0, R) is superstable whenever R is a regular sequence. We provide two proofs of this result. One relies on a result of E. Casanovas and M. Ziegler and the other on a quantifier elimination result. Both proofs share an important ingredient: for a regular sequence, the set of solutions of homogeneous equations is either finite or controlled by (finitely many) recurrence relations. This property resembles the Mann property defined by L. van den Dries and A. Günaydın in their work on expansions of fields by subgroups. Inspired by their work and the proofs of our main result, we show that when M is the domain of a multiplicative monoid of integers with the Mann property, then Z M is also superstable.

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“…For instance, a Theorem of Tao (see[20]) states that every natural number greater than 1 is the sum of at most five prime numbers.…”

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confidence: 99%

On expansions of (Z,+,0)

Lambotte

Point

2020

Annals of Pure and Applied Logic

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“…It is known that up to a given number x at most O(x 0.879 ) even integers do not have a Goldbach partition [30], and that every large enough even number is the sum of a prime and the product of at most two primes [24]. Furthermore, according to [48], every odd number greater that one is the sum of at most five primes. As described in Table 1, over a time span of more than a century the even Goldbach conjecture was confirmed to be true up to ever-increasing upper limits.…”

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confidence: 99%

Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸

2013

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This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than 4 • 10 18. Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to 8.37 • 10 26. The empirical data collected during this extensive verification effort, namely, counts and first occurrences of so-called minimal Goldbach partitions with a given smallest prime and of gaps between consecutive primes with a given even gap, are used to test several conjectured formulas related to prime numbers. In particular, the counts of minimal Goldbach partitions and of prime gaps are in excellent accord with the predictions made using the prime k-tuple conjecture of Hardy and Littlewood (with an error that appears to be O(√ t log log t), where t is the true value of the quantity being estimated). Prime gap moments also show excellent agreement with a generalization of a conjecture made in 1982 by Heath-Brown.

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“…Using the Hardy-Littlewood circle method, Vinogradov [24] established his famous theorem that every sufficiently large odd integer is the sum of three prime numbers. Effective versions of Vinogradov's theorem have been given by several authors (see [10,17,23] and references therein), but for the purposes of the present paper we require only the following extension of Vinogradov's theorem, which is due to Haselgrove [9, Theorem A]. 4.4.…”

Section: P Tmentioning

confidence: 99%

Zeta functions and asymptotic additive bases with some unusual sets of primes

Banks

2016

Ramanujan J

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A. Fix δ P p0, 1s, σ 0 P r0, 1q and a real-valued function εpxq for which lim xÑ8 εpxq ď 0. For every set of primes P whose counting function π P pxq satisfies an estimate of the formwe define a zeta function ζ P psq that is closely related to the Riemann zeta function ζpsq. For σ 0 ď 1 2 , we show that the Riemann hypothesis is equivalent to the non-vanishing of ζ P psq in the region tσ ą 1 2 u. For every set of primes P that contains the prime 2 and whose counting function satisfies an estimate of the form π P pxq " δ πpxq`O`plog log xq εpxq˘, we show that P is an exact asymptotic additive basis for N, i.e., for some integer h " hpPq ą 0 the sumset hP contains all but finitely many natural numbers. For example, an exact asymptotic additive basis for N is provided by the set t2, 547, 1229, 1993, 2749, 3581, 4421, 5281 . . .u, which consists of 2 and every hundredth prime thereafter.

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Every odd number greater than $1$ is the sum of at most five primes

Cited by 24 publications

References 39 publications

On expansions of (Z,+,0)

On expansions of (Z,+,0)

Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸

Zeta functions and asymptotic additive bases with some unusual sets of primes

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