264 History of the Theory of Numbers. [Chap, ix 
R. D. CarmichaeP" treated the problem to find m, given the prime p 
and s = ^ai, in Legendre's formula; a given solution m-, leads to an infinitude 
of solutions m-zv'', k arbitrary. Again, to find 771 such that p"*~' is the highest 
power of p>2 dividing m\, we have m — t={m — s)/{p — l), and see that m 
has a hmited number of values; there is always at least one solution m. 
Carmichael" used the notation H\y\ for the index of the highest power 
of the prime p dividing y, and evaluated 
;i=i/{n(xa+c)|, 
where a, c are relatively prime positive integers. Set Co = c and let % be 
the least integer such that iVa+Cr-i is divisible by p, the quotient being Cr. 
Let 
ei 
= [^4^']' ^'=[^]' ^>i- 
t-i 
Then /i= 2(6^+1), where t is the least subscript for which 
r=l 
Ct{a-\-Ct){2a-^Ct) . . . (cta+Ct) 
is not divisible by p. It follows that 
where R is the index of the highest power of p not exceeding 7i — 1 . If n is a 
power of p, /i = (n — l)/(p — 1). But if n = 8kp''-\- . . .-\-dip-\-8o, 8k9^0, and 
at least one further 8 is not zero, 
^SftSfc+^. <r = 6.+ ...+6o. 
P — 1 p — 1 
In case the first x for which xa-\-c is divisible by p gives c as the quotient, 
all the Cr are equal and hence all the v; then 
, _ rn — 1—i+pl . rn — l—i — ip-\-p^l , rn — l—i — ip — ip'^-i-p^l . 
L p J"^L p' J"^L w J"^'" 
The case a = c = l yields Legendre's^ result. The case a = 2, c = 1, gives 
Hll.3.5. . .(2„-l)( = [?^^] + [?^^^>. . .. 
E. Stridsberg^^ wrote H^ for (1) and considered 
Trt = a{a-\-m) . . .{a-{-'mt), 
where a is any integer not divisible by the positive integer m. Let p be a 
prime not dividing m. Write a^ for the residue of aj modulo m. He noted 
that, if pj=l (mod m), 
"BuU. Amer. Math. Soc, 14, 1907-8, 74-77; Amer. Math. Monthly, 15, 1908, 15-17. 
''Ibid., 15, 1908-9, 217. 
"Arkiv for Matematik, Astr., Fysik, 6, 1911, No. 34. 
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