440 HISTORY OF THE THEORY OF NUMBERS. [CHAP. XIV 



H. S. Vandiver, Amer. Math. Monthly, 9, 1902, 79-80; others, 7, 1900, 82-3, 112-3. 



A. Gerardin, Sphinx-Oedipe, 1906-7, 95, 161-2 [Vieta, 3 bibliography]. 



F. Ferrari, Suppl. al Periodico di Mat., 11, 1908, 77-8 [Frenicle 5 ]. 



A. Ge'rardin, Assoc. franc., 1908, 15-17 [bibliography]. 



A. Tafelmacher, 1'interme'diaire des math., 15, 1908, 102, 259. 



Welsch, ibid., 16, 1909, 19, 156 [no novelty in authors cited]. 



A. Martin, Amer. Math. Monthly, 25, 1918, 124. 



E. Bahier, Recherche . . . Triangles Rectangles en Nombres Entiers, 1916, 212-7. 



FOUR SQUARES IN ARITHMETICAL PROGRESSION. 



Fermat 41 proposed the problem to Frenicle May (?), 1640 and stated 

 (Fermat 11 of Ch. XV) that it is impossible. Euler 109 of Ch. XXII, P. 

 Barlow, 42 and M. Collins 43 proved the problem is impossible. 



B. Bronwin and J. Furnass 43a took relatively prime squares z 2 , y 2 , z 2 , w 2 . 

 By y 2 x 2 =z 2 y 2 = w 2 z 2 , we must have y+x = 2ab, yx = 2cd, z+y = 2ac, 

 zy = 2bd, w+z = 2bc, wz = 2ad. By the two values of y and those of z, 

 (a+d)b=(a d)c, (c+d}a = b(cd). But the g.c.d. of the four numbers 

 ad, cd is 1 or 2. Hence a+d = dc, ad = 5b, c+d = eb, cd = ta, 5 = 1 

 or 2, e = 1 or 2. These are inconsistent since a is prime to d. 



A. Genocchi 44 proved the impossibility of 4 squares in A. P. and the 

 following generalization (of the case p = 2) . The four expressions x=F(p+l)y 

 and x^F(pl)y are not all squares if p is a prime 8m3 such that p+1 and 

 p l admit no prime divisor 4m +1, and x, y are relatively prime. 



Several writers 45 failed to find a solution. 



L. Aubry 46 proved by descent the impossibility of 4 squares in A. P. 



E. Turriere 47 gave a proof. 



NUMBERS IN ARITHMETICAL PROGRESSION ALL BUT ONE BEING SQUARES. 



A. Guibert 48 noted that if A 2 , B 2 , C, D 2 (all but C being squares) are 

 in A. P., they are the products by a square of a similar progression of odd 

 integers relatively prime by twos. From the conditions A 2 +C = 2B 2 , 

 B 2 +D 2 = 2C, eliminate C. Then D 2 = 3B 2 -2A 2 . The known method of 

 solution gives 



A = 2p 2 -2pq-q 2 , b = 2p 2 +q 2 , d = 2p 2 +4pq-q 2 . 



A. Cunningham 49 found five integers in A. P., four being squares. If 

 v 2 , w 2 , X, y 2 , z 2 are in A. P., v 2 +3z 2 = (2y} 2 , 3v 2 +z 2 =(2iv} 2 , which require that 

 the five numbers be equal (Collins, 43 pp. 17-23) . Next, let all but the fourth 



41 Oeuvres, II, 195. 



42 Theory of Numbers, 1811, 257. 



43 A Tract on the possible and impossible cases of quadratic duplicate equalities . . . , Dublin, 



1858, 16. Abstract in British Assoc. Reports for 1855, 1856, Trans, of Sections, 4. 

 The Ladies' and Gentleman's Diary, London, 1857, 92-6. 

 430 The Gentleman's Diary, or Math. Repository, London, No. 73, 1813, 42-43. 



44 Comptes Rendus Paris, 78, 1874, 433-5. 



45 Amer. Math. Monthly, 5, 1898, 180. 

 48 Sphinx-Oedipe, 6, 1911, 1-2. 



47 L'enseignement math., 19, 1917, 240-1. 



48 Nouv. Ann. Math., (2), 1, 1862, 249-252. Cf. Pocklington 83 of Ch. 



49 Math. Quest. Educ. Times, (2), 9, 1906, 107-8. 



