By John G. Kemeny

Advent to Finite arithmetic

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Zhu YH, Yang BC. Improvement on Euler’s summation formula and some inequalities on sums of powers. Acta Scientiarum Naturalium Universitis Sunyatseni, 1997, 36(4): 21-26. 15. Xie ZT. A generalization of Stirling formula. Mathematical Practice and Cognition, 2006, 36 (6): 331-333. 16. Yang BC. Some new inequalities on step multiply. Journal of Guangdong Education Institute, 2002, 22(2):1-4. 17. Wei SR, Yang BC. An inequality of Stieltjes coefficients and estimation of their order. Journal of Central Nation University ( Natural Science), 1996,5(2):149~152.

2 n ( ne ) n e 12 n (1 1 ) 30 n2 1 2 n ( ne ) n e 24 . 35), it follows that | n | 4 e1/ 24 n ( en ) n , n 2 . 4. REFERENCES 1. Xu LZ, WANG XH. Methods on mathematical analysis and examples. Beijing: Higher Education Press, 1985. 2. Theory and application of infinite series. Londen: Blackie & Son Limited,1928. 3. Qu WL. Combination mathematics. Beijing: Beijing University Press, 1989. 4. Yang BC. A new formula for evaluating the sum of d-th powers of the first n terms of an arithmetic sequence.

6 If k ( x, y ) m 1 1 m y y 1 1 n ( , then it follows n 1 1 1 1 u u u du sin( / r ) 0 . 4 If k ( x, y ) ( ) dy 1 r , n 1 m 1 1 u r du rs 0 . 2 SOME EXAMPLES 1 r , then it follows n 1 p m 1 0 m y y 1 max{ x , y } amp . 27). By the same way, we can show the best possible property of the constant factor in the case of the reverse equivalent inequalities. 27). k (r ) ) [ p sin( / r ) (r , m) max{1m,n} ( mn ) amp . 28) ( ) r dy 1 ( ) du 0 , ln u 1 r 1 u 1 u ln u 1 1 u 1 u 0 l (r ) 1 ( ) s du ( s, n) ln(mm/nn ) ( mn ) s [ sin( / r ) ]2 .