22 HISTORY OF THE THEORY OF NUMBERS. [CHAP. I 



where e = 8 in the former case and e = 16 in the present case. For ex- 

 ample, 1, 210, 40755 are both triangular and pentagonal, whereas Barlow 90 

 stated that this is true only for unity. 



Gill 108 found n-gonal numbers whose sum and difference are n-gonal, 

 i. e., P x + P v = P 2 , P x - P y = P v , where P x = (n - 2)z 2 - (n - 4)z. 

 As a generalization, take P x = mx 2 m'x, where m and m' are relatively 

 prime. The first condition is satisfied if 



b f A f ^ f a 



z y = - (mx m), m(z + y) m = r. 



CL t/ 



Each of these linear equations is solved separately and the resulting re's 

 equated. The second of our conditions is treated similarly and the two 

 sets of values of x and y are compared. But the resulting solution does 

 not lead to "convenient numbers." Another method is to assume that 

 x = aw h, y = bw, z = cw h, where 



a 2 + 6 2 = c 2 , 2mh(c - a) = m'(a + b - c), 

 whence our first condition is satisfied. Thus take 



a = 2kl, b = k 2 - I 2 , c = k 2 + I 2 , I = mh, k = mh + m'. 

 The second of our initial conditions now becomes 



4w 2 (d 2 - 2m ft )w 2 - 4m(2mh + m')dw + (2mh + m') 2 = (2mv - m') 2 , 



where d = a m'\ Take 2mv m' = 2wt/u + 2mh + in'. We get w 

 and then v rationally. By choice of the denominator t 2 (d 2 2m' 4 ) w 2 ^ 2 

 we get integral answers unless m' = 0. 



Many 109 found two squares x 2 , y 2 such that x 2 y 2 are pentagonal. 

 Let 24(z 2 - y 2 ) + 1 = |4(z + 2/) 1 } 2 , whence x = 5y T 1. Then 



24(z 2 + y 2 ) + 1 = 624y 2 =F 240y + 25 = (5 - yr/s} 2 



determines y. Again, to find pentagonal numbers p, q whose sum and 

 difference are squares x 2 , y 2 , take 12(z 2 y 2 ) + 1 = {3 (a: + y) I} 2 and 

 12(z 2 + y 2 ) + 1 = (7 - yr/s) 2 , whence x = 7y 2. 



0. Terquem 110 proved that no triangular number > 1 is a biquadrate. 



The ordinary definitions of polygonal and figurate numbers as sums of 

 series were repeated by F. Stegmann, 111 George Peacock, 112 A. Transon, 113 

 H. F. Th. Ludwig, 114 Albert Dilling, 115 and V. A. Lebesgue. 116 



F. Pollock 117 stated that every integer is a sum of at most 10 odd squares, 

 and a sum of at most 11 triangular numbers 1, 10, 28, 55, of rank 3n+l, 



108 Math. Miscellany, Flushing, N. Y., 1, 1836, 225-230. 



109 The Lady's and Gentleman's Diary, London, 1842, 41-3, Quest. 1677. 



110 Nouv. Ann. Math., 5, 1846, 70-78. 



111 Archiv Math. Phys., 5, 1844, 82-89. 



m Encyclopaedia Metropolitana, London, 1, 1845, 422. 

 U3 Nouv. Ann. Math., 9, 1850, 257-9. 



114 Ueber fig. Zahlen u. arith. Reihen, Progr. Chemnitz, Leipzig, 1853. 



115 Die Progressionen, fig. u. polyg. Z., Progr. Muehlhausen, 1855. 

 114 Exercices d'analyse nume'rique, 1859, 17-20. 



117 Proc. Roy. Soc. London, 5, 1851, 922-4. Cf. Euler,' 8 - 7a Beguelin. 72 



