tailieunhanh - Báo cáo toán học: "The Last Digit of 2n n and n 2n−2i i n−i"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: The Last Digit of 2n n and n 2n−2i i n−i. | The Last Digit of 2n and EO 2n_2i n1 i n_i Walter Shur 11 Middle Road Port Washington NY 11050 wshur @ Submitted June 28 1996 Accepted November 11 1996. AMS subject classification 1991 Primary 05A10 Secondary 11B65. Abstract X X Let fn Ẹ 7 2nZi2i gn Ẹ O 22Z2T Let ak k i be the set of all positive integers n in increasing order for which 2 is not divisible by 5 and let bk k 1 be the set of all positive integers n in increasing order for which gn is not divisible by 5. This note finds simple formulas for ak bk 2 mod 10 fn mod 10 and gn mod 10. Definitions _ X n 2n-2i _ X n 2n-2i fn 2_i i n-i 1 g n M vlln-il ak k 1 is the set of all positive integers n in increasing order for which 2nn is not divisible by 5. bk k i is the set of all positive integers n in increasing order for which gn is not divisible by 5. un is the number of unit digits in the base 5 representation of n . THE ELECTRONIC .JOURNAL OF COmBINATORICS 4 no. 2 R16 2 Theorem 1. ak is the number in base 5 whose digits represent the number k in base 3. If n 1 mod 10 0 if n 2 akg 2 1 1 1 4 I. _r. I 2 6 if n 2 ak g and un mod 4 0 8 1 I 3 Note that if n 2 akg un is odd even if and only if n is odd even . Proof. From Lucas theorem 1 we have mod 5 where 2n Nr N3N2Ni 5 n ns n3n2ni 5 and t min r s . Suppose that for each i t m 2. Then for each i t Ni 2m. Since m 0 1 or 2 each term of the product is 1 2 or 6. Hence 2 is not divisible by 5. Suppose that for some i ni 2. Let im be the smallest value of i for which that is true. Then if nim is 3 or 4 Nim is 1 or 3 resp. . In either case T ị 0 and ị2 ỉ is divisible by 5. Thus ak g is the set of all positive integers written in base 3 but interpreted as if they were written in base 5. Since akg is in increasing order the hrst part of the theorem is proved. Suppose now that 2 is not divisible by 5. Then each term of the product Nt nt is 1 2 or 6 according as ni 0 1 or 2 . We have noting that 2 is even 2un mod 10 6y 2 4 or 8 N1 N2 No mod 10 6 2 4 or 8 nif .