210 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Or

nt+m+y+ys+ - +y.) =[m+ D" —mr](gn+ 1)
By the third formula

(n—D[m + 1) (gn+ 1) — 1]

={m+Hlm +1)"Xgn+1) = 1]
—(m+1)"(gn+1) + (m + 1)

nz

Hence
nt+it+yetys+ o +y.) Fnz=—m(gn+1)+m+1,

which equals the value of x given in the first formula, as it
should.

493. The value of x may be written in another form that
may be useful.

— m"® — n + 1 (mod m"t)

—mrH —r — 1

m(—m—1) — (n — 1)
=n—-1)Mm —1)

Hence x=(n—1)(m"* — 1) (mod m"+)

Therfor there exists an integer # such that

We have «x

I

x=rmt 4+ n—1)(m" —1)

494. Let us now consider special cases.

First let n = 5, whence m = — 5.
Then x—1=¢-5%+4+5—-35=g¢g-5° + 3120
14m=14+x—-1)5=¢5 + 625

14y = (1+y)4/5) =q-4-5* + 500
1+ys=(1+3)4/5) =g-4*5+ 400
14y, = g-43-52 4+ 320
14 95 = g-445 + 256