79.62 A Large Pair of Twin Primes

The relations between the kinetic energy spectrum and the second order longitudinal structure function in two dimensions are derived, and several examples are considered. The forward conversion (from spectrum to structure function) is illustrated first with idealized power law spectra, representing turbulent inertial ranges. The forward conversion is also applied to the zonal kinetic energy spectrum of Nastrom and Gage (1985) and the result agrees well with the longitudinal structure function of Lindborg (1999). The inverse conversion (from structure function to spectrum) is tested with data from 2D turbulence simulations. When applied to the theoretical structure function (derived from the forward conversion of the spectrum), the result closely resembles the original spectrum, except at the largest wavenumbers. However the inverse conversion is much less successful when applied to the structure function obtained from pairs of particles in the flow. This is because the inverse conversion favors large pair separations, which are typically noisy with particle data. Fitting the structure function to a polynomial improves the result, but not sufficiently to distinguish the correct inertial range dependencies. Furthermore the inversion of non-local spectra is largely unsuccessful. Thus it appears that focusing on structure functions with Lagrangian data is preferable to estimating spectra.

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An original abstract over the twin primes, the Goldbach conjecture, the friendly numbers, the perfect numbers, the Mersenne composite numbers, and the Sophie Germain primes

Journal of Discrete Mathematical Sciences and Cryptography ◽

10.1080/09720529.2008.10698400 ◽

2008 ◽

Vol 11 (6) ◽

pp. 715-726 ◽

Cited By ~ 2

Author(s):

Ikorong Anouk Gilbert Nemron

Keyword(s):

Twin Primes ◽

Perfect Numbers

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A Property of Twin Primes

Integers ◽

10.1515/integers-2011-0121 ◽

2012 ◽

Vol 12 (4) ◽

Cited By ~ 1

Author(s):

Christian Aebi ◽

Grant Cairns

Keyword(s):

Twin Primes ◽

Quadratic Residues

Abstract.We determine the product of the invertible quadratic residues in

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Magic squares of squares over a finite field

Topology, Geometry, and Dynamics - Contemporary Mathematics ◽

10.1090/conm/773/15536 ◽

2021 ◽

pp. 111-122

Author(s):

Stewart Hengeveld ◽

Giancarlo Labruna ◽

Aihua Li

Keyword(s):

Finite Field ◽

Prime Numbers ◽

Similar Question ◽

Magic Squares ◽

Content Type ◽

Magic Square ◽

Twin Primes ◽

Quadratic Residues ◽

Perfect Squares ◽

Open Question

A magic square M M over an integral domain D D is a 3 × 3 3\times 3 matrix with entries from D D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M M are perfect squares in D D , we call M M a magic square of squares over D D . In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring Z \mathbb {Z} of the integers which has all the nine entries distinct?” We approach to answering a similar question when D D is a finite field. We claim that for any odd prime p p , a magic square over Z p \mathbb Z_p can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p p such that, over Z p \mathbb Z_p , magic squares of squares with nine distinct elements exist. In addition, if p ≡ 1 ( mod 120 ) p\equiv 1\pmod {120} , there exist magic squares of squares over Z p \mathbb Z_p that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

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