Restricted linear congruences

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• 作者：Khodakhast Bibak^a ; ^{kbibak@uvic.ca" class="auth_mail" title="E-mail the corresponding author} ; Bruce M. Kapron^a ; ^{bmkapron@uvic.ca" class="auth_mail" title="E-mail the corresponding author} ; Venkatesh Srinivasan^a ; ^b ; ^{srinivas@uvic.ca" class="auth_mail" title="E-mail the corresponding author} ; Roberto Tauraso^c ; ^{tauraso@mat.uniroma2.it" class="auth_mail" title="E-mail the corresponding author} ; Lá ; szló ; Tó ; th^d ; ^{ltoth@gamma.ttk.pte.hu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11D79 ; 11P83 ; 11L03 ; 11A25 ; 42A16
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：171
• 期：Complete
• 页码：128-144
• 全文大小：414 K

文摘

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence md5=3d7accee7ffdc07316f2a1f95c6fc05d">$a_1{x}_1+\cdots +a_k{x}_k\equiv b(\mathrm{mod}n)$, with md5=a88d9f985ecd3e3a9d99522d3dbd70e0" title="Click to view the MathML source">gcd⁡(x_i,n)=t_i$\gcd (x_i,n)=t_i$ (md5=336bb509162c0d3a9ac48daf933e13be" title="Click to view the MathML source">1≤i≤k$1\le i\le k$), where md5=e1b405fde4fda511ea9382deb41a36c9" title="Click to view the MathML source">a₁,t₁,…,a_k,t_k,b,n$a_1,t_1,\dots ,a_k,t_k,b,n$ (md5=aa6305b31f5c4009b2fd05ef6a0ac9ec" title="Click to view the MathML source">n≥1$n\ge 1$) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of md5=c1809322cbfb0fdbccd5baec4e622656" title="Click to view the MathML source">a_i=t_i=1$a_i=t_i=1$md5=0547e290b77712403c266d9d64f2a3ac" title="Click to view the MathML source">(1≤i≤k)$(1\le i\le k)$. Since then, this problem has been studied, in several other special cases, in many papers; in particular, Jacobson and Williams [Duke Math. J.   39 (1972) 521–527] gave a nice explicit formula for the number of such solutions when md5=fbb1cce0eb40e3f4b7a65d967de9023f" title="Click to view the MathML source">(a₁,…,a_k)=t_i=1$(a_1,\dots ,a_k)=t_i=1$md5=0547e290b77712403c266d9d64f2a3ac" title="Click to view the MathML source">(1≤i≤k)$(1\le i\le k)$. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.