"L'interm^diaire des math., 20, 1913, 42-44. 
^^Ibid., 80. 
'*Ibid., 124, 262, 283-4. 
"Atti Accad. Romana Nuovi Lincei, 66, 1912-3, 43-5. 
"Archiv Math. Phys., (3), 22, 1914, 365-6. 
"Giomale di Mat., 52, 1914, 53-7. 
'"L'interm^diaire des math., 21, 1914, 23^, 58. 
"/6id., 22, 1915, 110-1. Objections by MaUlet, 23, 1916, 10-12. 
"Suppl. al Periodico di Mat., 19, 1915, 17-23. 
"''Sphinx-Oedipe, 9, 1914, 42. 
"Sitzungsber. Berhn Math. Gesell., 15, 1915, 8-18. 
"Nouv. Ann. Math., (4), 17, 1917, 234. 
"L'intermddiaire des math., 24, 1917, 31-2. 
»/Wd., 96. Cf. H. Brocard, 25, 1918, 35-8, 112-3. 
I 
464 History of the Theory of Numbers. [Chap, xx 
A. Cunningham"^ listed 63 symmetrical numbers aoaia2aiao each a 
product of two symmetrical numbers of 3 digits, and all numbers n^, 
r2< 10000, and all n'\ n\ n^, n^\ 7i<1000, ending with 2, 7, 8, symmetrical 
with respect to 2 or 3 digits, as 618' = 236029032. f ie^' 
Pairs'^ of numbers whose 1. c. m. equals the product of the digits. 
Pairs'^ of biquadrates, cubes and squares having the same digits. 
*P. de Sanctis'^ noted a property of numbers to the base h^+1. 
L. von Schrutka^^ noted that 15, 18, 45 in 7-15 = 105, 6-18 = 108 and 
9-45 = 405 are the only numbers of two digits which by the insertion of 
zero become multiples. 
G. Andreoli^^ considered numbers N of n digits to the base k whose rth 
powers end with the same ?i digits as N. Each decomposition of k into 
two relatively prime factors gives at most two such N's. If the base is a 
power of a prime, there is no number > 1 whose square ends with the same 
digits. 
Welsch^° discussed the final digits of pth powers. 
H. Brocard^ ^ discussed various powers of a number with the same sum 
of digits. _ 
A. Agronomof®^ wrote N for the number obtained by reversing the digits 
of N to base 10 and gave several long formulas for 2ji J j. 
The^-'' only case in which N^—N"^ is a square for two digits is 65- — 
56^=33". There is ^o case for three digits. 
R. Burg^ found the numbers N to base 10 such that the number 
obtained by reversing its digits is a multiple kN of A^, in particular for 
A: = 9, 4. 
E. Lemoine^ asked a question on sjinmetrical numbers to base b. 
H. Sebban^^ 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. Goormaghtigh^^ noted that this property of the squares of 5, 6 and 
45, 56 is a special case of A^—B"^ — !. . .1 (to 2p digits), where A = 5. . .56, 
5 = 4. . .45 (to p digits). Again, the factorizations 11111=41-271, 1111111 
= 239-4649 yield the answers 115^, 156^ and 2205^, 2444^. 
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