no. 9-10
Combinatorics/Number Theory
Partition regularity and the primes
[Régularité de partitions et les nombres premiers]

Présenté par : Jean-Pierre Kahane

Lê, Thái Hoàng1
1 Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, TX 78712, USA
Comptes Rendus. Mathématique, Tome 350 (2012) no. 9-10, pp. 439-441.

Dans cette Note, on montre, comme une conséquence simple du programme de Green et Tao sur le comptage de configurations linéaires dans les nombres premiers et du travail de Deuber sur la régularité de partitions, que si un système dʼéquations est régulier par rapport aux partitions des nombres entiers, alors il est régulier par rapport aux partitions des ensembles {p−1:p premier} ainsi que {p+1:p premier}. Cela répond à une question de Li et de Pan.

The purpose of this Note is to point out, as a simple yet nice consequence of Green and Taoʼs program on counting linear patterns in the primes and Deuberʼs work on partition regularity, that if a system of equations is partition regular over the positive integers, then it is also partition regular over the sets {p−1:p prime} as well as {p+1:p prime}. This answers a question of Li and Pan.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.04.011
Lê, Thái Hoàng 1

1 Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, TX 78712, USA 
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Lê, Thái Hoàng. Partition regularity and the primes. Comptes Rendus. Mathématique, Tome 350 (2012) no. 9-10, pp. 439-441. doi : 10.1016/j.crma.2012.04.011. http://www.numdam.org/articles/10.1016/j.crma.2012.04.011/

[1] Bergelson, V.; Leibman, A.; Ziegler, T. The shifted primes and the multidimensional Szemeredi and polynomial van der Waerden theorems, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 3–4, pp. 123-125

[2] Brauer, A. Über Sequenzen von Potenzresten, Sitzungsberichte der Preußischen, Akademie der Wissenschaften, Mathematisch-Physikalische Klasse, 1928 (pp. 9–16)

[3] D. Conlon, W.T. Gowers, Combinatorial theorems in sparse random sets, preprint.

[4] Deuber, W. Partitionen und lineare Gleichungssysteme, Math. Z., Volume 133 (1973), pp. 109-123

[5] Green, B. Rothʼs Theorem in the primes, Ann. of Math., Volume 161 (2005) no. 3, pp. 1609-1636

[6] Green, B.; Tao, T. The primes contains arbitrarily long arithmetic progressions, Ann. of Math., Volume 167 (2008), pp. 481-547

[7] Green, B.; Tao, T. Linear equations in primes, Ann. of Math., Volume 171 (2010) no. 3, pp. 1753-1850

[8] Green, B.; Tao, T. The Möbius function is strongly orthogonal to nilsequences, Ann. of Math., Volume 175 (2012) no. 2, pp. 541-566

[9] B. Green, T. Tao, T. Ziegler, An inverse theorem for the Gowers Us+1[N]-norm, Ann. of Math., in press.

[10] Hindman, N. Partition regularity of matrices (Landman, B.; Nathanson, M.; Nesetril, J.; Nowakowski, R.; Pomerance, C., eds.), Combinatorial Number Theory, de Gruyter, Berlin, 2007, pp. 265-298

[11] Hindman, N.; Leader, I. Image partition regularity of matrices, Combin. Probab. Comput., Volume 2 (1993), pp. 437-463

[12] Li, H.; Pan, H. A Schur-type addition theorem for primes, J. Number Theory, Volume 132 (2012) no. 1, pp. 117-126

[13] Rado, R. Studien zur Kombinatorik, Math. Z., Volume 36 (1933), pp. 242-280

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