[3]

If we have three equations, the equations of condition that give the values of the quantities b, may be obtained in a similar manner from the first three equations of the systems (1), (2), (3), &c.

(G. A. 25; Bezout, Theor. des Equ. Alg. 195~-223.)

(57.) Every square number is of the same form, with regard to the modulus 2a, or 2a+1, or 4a, as one of the

0°, 12, 2o, 3o, &c. a2.

squares

Table of the possible Forms of square Numbers for every

[1] a, and b, may be supposed not to contain a square factor; [2] x, y, and x, may be considered prime to each other;

[3] if the equation is impossible in integers, it is also impossible in fractions.

If ma+b is a possible form, and ma+c an impossible form, of square numbers to the modulus a, the equation

(ma+b) x2+nay2 = x2

is always possible; and the equation

(ma + c) x2+nay2 = x2

is always impossible, if n is prime to a.

K

Table of the Remainders of Squares to every Modulus, not containing a square Factor, from 2 to 51.

(B. p. 104.)

37

38

39

41

1 3 4

1 2 4

9 12 15 16 22 25 27 31

8 9 13 15 16 17 18 19 21 25 26 30 32 33

1 4 9 11

1 3 4

14 15 16 21 25 29 30

7 9 10 11 12 16 21 25 26 27 28 30 33 34 36

145 6 7 9 11 16 17 19 20 23 24 25 26 28 30 35 36

1 3 4 9 10 12 13 16 22 25 27 30 36

124 5 8 9 10 16 18 20 21 23 25 31 32 33 36 37 39 40

147 9 15 16 18 21 22 25 28 30 36 37 39

42

43

146 9 10 11 13 14 15 16 17 21 23 24 25 31 35 36 38 40 41

46

1 2 3 4 6 8 9 12 13 16 18 23 24 25 26 27 29 31 32 35 36 39 41

47

1 2 3 4 6 7 8 9 12 14 16 17 18 21 24 25 27 28 32 34 36 37 42

51

1 4 9 13 15 16 18 19 21 23 25 30 34 36 42 43 49

Let a be a prime number, and b prime to a; then if

be severally divided by a, the remainders will all be unequal.

The product of a possible and impossible form of squares to the same modulus is always impossible. (B. 42-52.)

Conditions of the possibility of the equation

x2 — a y2 = b x2,

in which a and b are positive integers, and a < b.

Let a number c be found such that c>b, and c2— a÷b;

C

and let the following system of quantities be constructed:

then if a, b, b,, are such that any integral values of e,, C2, e1, e, will satisfy the conditions

in which a, b, and c are prime to each other, is possible, if the conditions ae2+b2÷c, ce22-b÷a, ce2 — a÷b,

2

2

may be fulfilled by any integral values of e1, eg, ez.

Hence may be derived the following rule: divide b and c by a, then if both or neither of the remainders are found

amongst the remainders of squares to the modulus a, the equation may be possible. If this condition is fulfilled, and the same relation exists between the remainders of a, and c, when divided by b, and those of a, and c-b, when divided by c, the equation is possible; if either of these three conditions. fails, the equation is impossible. (B. 53, 178; Leg. 23-7.)

Impossible pairs of quadratic equations:

EQUATIONS OF THE SECOND DEGREE.

(58.) Solution of particular equations.

Let the given equation be a2+1=y2,