492 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. XVIII

Factoring the difference, we set a(c— b}n=A — B, 2dbc+bcn~A+B. Insert the resulting value of A into the equation involving A2. We find that

{(ac+bc-ab)2-4:abc2}2 ~

_

c— ab)(ac+ab— bc)(ac— ab— be)* The initial squares will be in A. P. if we take

a = 2rs-r2+s2, 6 = r2+s2, c = 2rs+r2-s2;

whence a=l, 6 = 5, c = 7 if r = 2, s = l. Then x = 15132 1/7863240, a result found by J. D. Williams6 by starting with the squares z2, 25s2, 49x2. For r=4, s = 3, we get a = 17, 6 = 25, c=31, z= -Z, where

X= (864571)2/H011044931800,

and hence a solution of a2X2-aX=D, • • •, c2X2-cZ=D [Perkins28 of Ch. XIV].

Hart7 made &2z2+/bz=D for k = a', V, ••-. Divide by &2 and set a = l/a', •••. Then z2-j-aa;, x2+6a;, ••• are to be squares. Set # = 22. Then z*+a, 2^+6, • • • are to be squares. Suppose that z2 is a sum of two squares in the required number of ways: 22 = m2+n2 = p2+g2= • • •, and set a = 2mn, l = 2pq, • • •. Then z2+a = (m+n)2, z2+& = (p+q)2,

J. Matteson8 gave the solutions by Hart5» 7 with amplifications.

G. B. M. Zerr9 solved the system x2 + y2 = z2 + w2 = D, x2-w*=z2~-y2= D, also the system

;= D, (w2-n2)2o;2db(m2-n2)a:= D,

P. von Schaewen10 made 4z2-22, 4x2+3x, 4x2+5x all squares. Setting a? = l/(4o?i), we are to make l~2o;i, l+3o:i, l+5o;i all squares [von Schaewen81 of Ch. XV].

On three squares which increased or decreased by their roots give squares, see papers 12, 12a, 21, 26, 52-54 of Ch. XIV. For two squares, papers 3, 19 of Ch. XVI; 32 of Ch. XVII.

THREE LINEAR AND QUADRATIC FUNCTIONS OF TWO UNKNOWNS MADE

SQUARES.

Brahmegupta2 of Ch. XV made x+y, x—y and xy+1 all squares.

To find two numbers whose product is a square and product plus the square of either is a square, J. Hampson11 took 62a and a as the numbers. It remains to make 62+l = D = (6— c)2, say, which gives b. R. Maliock

• Algebra, BostoD, 1840, 413. ~~~

T Math. Quest. Educ. Times, 39, 1883, 47-9.

8 Collection of Diophantine Problems with Solutions (ed., A. Martin), Washington, D. C.,

1888, pp. 10-20. 8 Amer. Math. Monthly, 15, 1908, 17-18. Erroneous solution in J. D. Williams' Algebra,

1832, 419.

10 Archiv Math. Phys., (3), 17, 1911, 249-250. " Ladies Diary, 1763, p. 34, Quesfc. 491; Leybourn's Math. Quest. L. D., 2, 1817, 209.