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*a, X1 + "Dzel, + łazilz + '&q&a='a,

Paj & , +%a2X2 + 2azXz + Q4X4=?a;
x, = 16+ (-az.'a, - *ag.'az) U2 + (-az.'04-az.) Uz

+ ('ag.*Q — *ag.'aa) Uz,
Xx=26– (*a,.az a7.-ag)u,- (-a.Pat— ?a.'a,)u,

+ ('az:*04 'ag. '02) U4,
W3=36 + ('a.'a, a,.'a,)U,-('a.a4 — ?a;.'a.) Uz

- ('az.-04 — *ag."0_) 04,
X4=*6-('a,.'a, – *a,.'a,)u,- (a.az-a7.4a3) Uz

- ('ag.*az — ?dz.az) 44, [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.

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

0%, 1%, 2, 3, &c. .

Table of the possible Forms of square Numbers for every

Modulus from 3 to 20.

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In investigating the possibility of the equation

a x? +byo = x, [1] a, and b, may be supposed not to contain a square factor; [2] x, y, and %, 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

( a +b)^+ ago: is always possible; and the equation

(m a + c)2*+may=x^ 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.)

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10

11

13

14

15

17

19

21

22

23

26

29

30

1 4 5 6 9
1 3 4 5 9
1 3 4 9 10 12
1 2 4 7 8 9 11
1 4 6 9 10
1 2 4 8 9 13 15 16
1 4 5 6 7 9 11 16 17
1 4 7 9 15 16 18
1 3 4 9 11 12 14 15 16 20
1 2 3 4 6 8 9 12 13 16 18
1 3 4 9 10 12 14 16 17 22 23 25
1 4 5 6 7 9 13 16 20 22 23 24 25 28
1 4 6 9 10 15 16 19 21 24 25
1 2 4 5 7 8 9 10 14 16 18 19 20 25 28
1 3 4 9 12 15 16 22 25 27 31
1 2 4 8 9 13 15 16 17 18 19 21 25 26 30 32 33
1 4 9 11 14 15 16 21 25 29 30
1 3 4 7 9 10 11 12 16 21 25 26 27 28 30 33 34 36
1 4 5 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
1 2 4 5 8 9 10 16 18 20 21 23 25 31 32 33 36 37 39 40
1 4 7 9 15 16 18 21 22 25 28 30 36 37 39
1 46 9 10 11 13 14 15 16 17 21 23 24 25 31 35 36 38 40 41
1 2 3 4 6 8 9 12 13 16 18 23 24 25 26 27 29 31 32 35 36 39 41
1 2 3 4 6 7 8 9 12 14 16 17 18 21 24 25 27 28 32 34 36 37 42
1 4 9 13 15 16 18 19 21 23 25 30 34 36 42 43 49

31

33

34

35

37

38

39

41

42

43

46 47 51

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

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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

X? ay=bzo,

in which a and b are positive integers, and a < b.
Let a number c be found such that c > {b, and cacob;
and let the following system of quantities be constructed:

c? - a=b b,%, c=m, b, c »lba;
c? - a=b, b2, Ca=m,b2+ ba;

0," – a=bbzzz', &c.= &c. then if a, b, b, are such that any integral values of cy, C2, en, en, will satisfy the conditions

carb, e' - ba;

c - acub, e, – b,a; the equation is always possible.

The equation ax+byø=cx, in which a, b, and c are prime to each other, is possible, if the conditions ae ? + bac, cege - hea, cez - acob, may be fulfilled by any integral values of ez, ez, 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. 237)

Impossible pairs of quadratic equations :

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