丢番图方程ax+by=n的一个注记（英文）

A Remark of Diophantine Equantion ax+by=n

令a,b为互素的正整数,n为非负整数.D(a,b;n)表示不定方程ax+by=n的非负整数解(x,y)的个数Tripathi证明了■,其中ζ_m=e^(2πi/m).在本文中,我们建立了D(a,b;n)的递推关系,从而给出了上述结论的新证明.

Let a, b be positive integers such that （a, b）= 1 and let n be a non-negative integer. Define D（a,b;n） to be the number of non-negative integer solutions（x ,y）of the Diophantine equation ax＋by=n. Tripathi proved that D（a,b;n）=n/ab＋1/2（1/a＋1/b）＋1/aj=1∑a-1ζa^-jn/1-ζa^bj＋1/bk=1∑b-1ζb^kn/1-ζb^ak, where ζm = e^2πi/m. In this note, we put forward a recurrence relation of D（a, b; n） , thus giving a new proof of above formula.