204 History of the Theory of Numbers. [Chap, vil 
A. Cunningham^^^ gave five primes p for which there is a maximum 
number of exponents to which the various numbers belong modulo p. 
On exponents and indices, see Lebesgue^"'*^ and Bouniakowsky^^^; also 
Reuschle^^ of Ch. YI, Bouniakowsky"^ of Ch. XIV, and Calvitti^^ of Ch. XX. 
Binomial Congruences. 
Bhdscara Achd,rya^*^ (1150 A. D.) found y such that y^ — SO is di\'isible 
by 7 by solving ?/" = 7c+30. Changing 30 by multiples of 7, we reach a 
perfect square 16 with the root 4. Hence set 
7c+30 = (7n+4)2, c = 7n'+8n-2, y = 7n-\-4. 
Taking n = 1, we get y = ll. Such a problem is impossible if, after abrading 
the absolute term (30 above) by the divisor (7 above) and the addition of 
multiples of the divisor, we do not reach a square. 
Similarly for the case of a cube, with corresponding conditions for impos- 
sibihty (§206, p. 265). For y^ = 5e+Q, abrade 6 by the divisor 5 to get 
the cube 1; adding 43-5, we get 216 = 6^. Hence set y = 5n-\-Q. 
An anonymous Japanese manuscript^^° of the first part of the eighteenth 
century gave a solution of x^ — ky = a by trial. The residues Oi, . . ., ak-\ of 
1", . . ., (^ — 1)" modulo k are formed; if a^^a, then x = r. It was noted 
that ak-r = 0'r or k—Qr according as k is even or odd, and that the residue 
of r" is r times that of r"~^ 
Matsunaga,^^*^" in the first half of the eighteenth century, solved 
a}-\-hx= y^ by expressing 6 as a product mn and finding p, q and A so that 
mp — nq=l, 2pa=A (mod n). Then x={Am — 2a)A/n [and y=a — 7nb]. 
But if Am= 2a, write A-\-n in place of A and proceed as before. Or write 
2a+h in the form bQ+R, whence x=2a+b-{Q+l)R. To solve 69+ 
llx=y'^, consider the successive squares until we reach 5^=3 (mod 11). 
Write 2-5+11 in the form 1-11 + 10. Then for a=5, 6= 11, Q= 1, i2= 10, 
the preceding expression for x becomes 1, whence 5^+11-1 = 6^. Then 
write 2-6+11 in the form 2-11 + 1. Then 23-(2+l)-l = 20 gives 6"+ 
20-11= 16^, and a;= (256-69)/ll= 17. 
L. Euler^^^ proved that, if n divides p — l, where p is a prime, and if 
a = c''-\-kp, then (by powering and using Fermat's theorem), a^^~^^^" — l is 
divisible by p. Conversely, if a'" — 1 is divisible by the prime p = w7i+l, 
we can find an integer y such that a — ?/" is divisible by p. For, 
o'"-2/'"'' = (a-2/")Q(^), 
and the differences of order mn—n of Q(l), Q(2),. . ., Q{mn) are the same 
>"Math. Quest, and Solutions (Ed. Times), 3, 1917, 61-2; corrections, p. 65. 
'"Vlja-ganita, §§ 204-5; Algebra, with arith. and mensuration, from the Sanscrit of Brahmegupta 
and Bhdscara, transl. by H. T. Colebrooke, London, 1817, pp. 263-4. 
""Abhand. Geschichte Math. Wiss., 30, 1912, 237. 
^'^Ibid., 234-5. 
i"Novi Comm. Acad. Petrop., 7, 1758-9 (1755), p. 49, eeq., §64, §72, §77; Comm. Arith., 1, 
270-1, 273. In Novi Comm., 1, 1747-8, p. 20; Comm. Arith., 1, p. 60, he proved the first 
statement and stated the converse 
