464 HISTORY or THE THEORY OF NUMBEBS. [CHAP, xx A. Cunningham74 listed 63 symmetrical numbers aoa^aiao each a .product of two symmetrical numbers of 3 digits, and all numbers n3, n< 10000, and all n5, n7, n9, n11, n<1000, ending with 2, 7, 8, symmetrical with respect to 2 or 3 digits, as 6183= 236029032. Pairs75 of numbers whose 1. c. m. equals the product of the digits. Pairs76 of biquadrates, cubes and squares having the same digits. *P. de Sanctis77 noted a property of numbers to the base h?+l. L. von Schrutka78 noted that 15, 18/45 in 745 = 105, 6-18 = 108 and 945=405 are the only numbers of two digits which by the insertion of zero become multiples. G. Andreoli79 considered numbers N of n digits to the base k whose rth powers end with the same n digits as'AT". Each decomposition of k into two relatively prime factors gives at most two such N'a. If the base is a power of a prime, there is no number >l whose square ends with the same digits. Welsch80 discussed the final digits of pth powers. H. Brocard81 discussed various powers of a number with the same sum of digits. __ A. Agronomof82 wrote N for the number obtained by reversing the digits of N to base 10 and gave several long^formulas for ^J2i J. The820 only case in which.N2—N2 is a square for two digits is 652 — 562=332. There is no case for three digits. R. Burg83 found the numbers N to base 10 such that the number obtained by reversing its digits is a multiple kN of N, in particular for fc = 9,4. E. Lemoine84 asked a question on symmetrical numbers to base 6. H. Sebban85 noted that 2025 is the only square of four digits which yields a square 3136 when each digit is increased by unity. Similarly, 25 is the only one of two digits. R. Goormaghtigh86 noted that this property of the squares of 5, 6 and 45, 56 is a special case of A2—B* = 1... 1 (to 2p digits), where A = 5...56, B=4.. .45 (to p digits). Again, the factorizations 11111 =41-271,1111111 =2394649 yield the answers 1152,1562 and 22052, 24442. 74L'intenn6diaire des math., 20, 1913, 42-44. nlbid., 80. w/6id., 124, 262, 283-4. "Atti Accad. Romana Nuovi Lincei, 66, 1912-3, 43-5. "Archiv Math. Phys., (3), 22, 1914, 365-6. "Giornale di Mat., 52, 1914, 53-7. 80L'interme*diaire des math., 21, 1914, 23-4, 58. «/6id., 22, 1915,110-1. Objections by Maillet, 23, 1916, 10-12. 82Suppl. al Periodico di Mat., 19, 1915, 17-23. ^Sphinx-Oedipe, 9, 1914, 42. "Sitzungsber. Berlin Math. GeselL, 15, 1915, 8-18. MNouv. Ann. Math., (4), 17, 1917, 234. ML'interme*diaire des math., 24, 1917, 31-2. "Ibid., 96. Cf. H. Brocard, 25, 1918, 35-8, 112-3.