证明了:假设λ1,...,λ6是正实数,λ1/λ2是无理数, Dirichlet L 函数满足黎曼猜想, x1,…,x6是正整数,那么,λ1x12+λ2x22+λ3x33+λ4x43+λ5x53+λ6x63的整数部分可表示无穷多素数.
Abstract
The present paper proved that if λ1,...,λ6 are positive real numbers, λ1/λ2 is irrational, all Dirichlet L-functions satisfy the Riemann Hypothesis, then, the integer parts of λ1x12+λ2x22+λ3x33+λ4x43+λ5x53+λ6x63are infinitely many primes for natural numbers x1,…,x6.
关键词
整数变量 /
丢番图逼近 /
Davenport--Heilbronn方法
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Key words
integer variables /
diophantine approximation /
Davenport-Heilbronn method
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参考文献
[1] Davenport H., Helbronn H., On indefine quadratic forms in five variables, J. London Math. Soc., 1946, 21:185-193.
[2] Watson G. L., On indefinite quadratic forms in five variables, Proc. London Math. Soc., 1953, 3: 170-182.
[3] Bambah R. P., Four squares and a kth power, Quart. J. Math. Oxford, 1954, 5: 191-202.
[4] Davenport H., Roth K. F., The solubility of certain diophantine inequalities, Mathematika, 1955, 2: 81-96.
[5] Cook R. J., Diophantine inequalities with mixed powers, II, J. Number Theory, 1979, 11 (1): 49-68.
[6] Brüdern J., Kawada K., Wooley T. D., Additive representation in thin sequences, VIII: Diophantine inequalitiesin review, Series on Number Theory and Its Applications, 2010, 6: 20-79.
[7] Vaughan R. C., A new iterative method in Waring’s problem, Acta Math., 1989, 162: 1-71.
[8] Brüdern J., Perelli A., The addition of primes and power, Can. J. Math., 1996, 48(3): 512-526.
[9] Vaughan R. C., The Hardy-Littlewood Method, Second Edition, Cambridge Tracts in Mathematics, Vol.125, Cambridge University Press, Cambridge, 1997.
[10] Brüdern J., Kawada K., Wooley T. D., Additive representation in thin sequences, I: Waring’s problem forcubes, Ann. Scient. Ec. Norm. Sup., 2001, 34: 471-501.
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脚注
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基金
国家自然科学基金资助项目(11071070);河南省教育厅自然科学研究计划(2011B110002)
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