CHAPTER XIV. 



SQUARES IN ARITHMETICAL OR GEOMETRICAL PROGRESSION. 



THREE SQUARES IN ARITHMETICAL PROGRESSION, 



This topic is closely connected with congruent numbers, Ch. XVI, 

 especially papers 41, 67, 68, 120, 141. It may be stated in terms of tri- 

 angular numbers (Ch. I 179 ). 



Diophantus, III, 9, used three special squares in A. P. (see Ch. XV). 



Jordanus Nemorarius 1 in the thirteenth century found that 



r = 6 2 -c 2 /2, v = b 2 +bc+c 2 /2, g = 6 2 +26c+c 2 /2 



make v 2 r 2 = q 2 v 2 = 26 3 c+36 2 c 2 +6c 3 . Here b is any integer, c any even 

 integer. In his notations, set a = 6+c, d = a+b, h = ac, k bc, e ad,f=bd. 

 Then e = h+k+f, and a solution is y=(/i+/)/2, r=fv, q = e v. 



Regiomontanus, 2 or Johann Miiller (1436-1476), proposed in letters the 

 problems: Find 3 squares in A. P., the sum of whose integral roots is 214; 

 find 3 squares in A. P., the least being > 20000; find 3 squares in harmonical 

 progression. 



F. Vieta 3 took A 2 , (A+) 2 and (D-A) 2 as the squares. Hence 



Hence we may take D 2 -2B 2 , D-+2B-+2BD and D 2 +2B 2 +4BD as the 

 sides of the three squares. 



Fermat 4 proposed to St. Croix, Sept., 1636, that he find three squares in 

 A. P. the common difference being a square. 



Fermat 5 knew a rule for finding three numbers whose squares are in 

 A. P. Apparently the numbers were r 2 2s 2 , r 2 +2rs+2s 2 , r 2 +4rs+2s 2 . 

 Replacing r by p q and s by q, we obtain Frenicle's set p 2 2pq q 2 , 

 P 2 +<? 2 , p z +2pq q 2 . To derive the latter, Frenicle 6 noted that the squares 

 of a b, c, a+b are in A. P. if a?-\-b 2 = c 2 , and took a = p 2 q 2 , b = 2pq, 

 c = p 2 -{-q 2 . 



L. Euler 7 deduced from the solution y = l of l-\-x 2 = 2y* the solution 

 y = 13, etc. Cf . Cunningham 79 of Ch. XX. 



To find three integers the sum of any two of which is a square and whose 

 squares are in A. P., " Amicus " 8 took 2a 2 6 2 (a 4 -6 4 ), a 4 +6 4 as the 



1 Elementa Arithmetica decem libris, demostrationibus Jacobi Fabri Stapulensis, Paris, 1496, 



1514, Book 6, Theorem 12. 

 2 C. T. de Murr, Memorabilia Bibl. publ. Norimbergensium, Para I, 1786, 145, 159, 201. 



Cf. M. Cantor, Geschichte der Math,, II, 1892, 241, 263. 



3 Zetetica, 1591, V, 2; Opera Math., 1646, 76. Same by J. Prestet, Siemens des Math., ou 



Principes . . . , Paris, 1675, 328. 



4 Oeuvres, II, 65; III, 287. 



6 Oeuvres, II, 234; letter from Frenicle to Fermat, Sept. 6, 1641 (tables by Frenicle, p. 237). 



6 Triangles rectangles en nombres, prop. XI. Full reference in Ch. IV. 62 



7 Algebra, 2, 1770, Ch. 9, Art. 140; French transl., 2, 1774, p. 167; Opera Omnia, (1), I, 402. 



8 Ladies' Diary, 1795, 38, Quest. 974; Leybourn's Math. Quest, from L. D., 3, 1817, 297. 



435 



