CHAPTER XX.
PROPERTIES OF THE DIGITS OF NUMBERS.
John Hill1 noted that 139854276= 118262 is formed of the nine digits permuted and believed erroneously that it is the only such square.
N. BrownelP found 169 and 961 as the squares whose three digits are in reverse order and whose roots are composed of the same digits in reverse order. The least digit in the roots is also the least in the squares, while the greatest digit in the roots is one-third of the greatest in the squares and one-half of the digit in the tens place.
W. Saint1* proved that every odd number N not divisible by 5 is a divisor of a number 11... 1 of D^ N digits [by a proof holding only for N prime also to 3]. For, let 1... 1 (to D digits) have the quotient q and remainder r when divided by D. This remainder r must recur if the number of digits 1 be increased sufficiently. Hence let 1...1 (to D+d digits) give the remainder r and quotient Q when divided by D. By subtraction, D(Q—q) = 1.. .10.. .0 (with d units followed by D zeros). Hence it 1.. .1 (to d digits) were not divisible by every odd number ^ D and prime to 5 [and to 3], there would be a remainder R] then #0.. .0 (with D zeros) would be divisible by an odd number prime to 5 [and to 3], which is impossible.
P. Barlow10 stated, and several gave inadequate proofs, that no square has all its digits alike. Held stated and proved that 1111111112= 123456-78987654321 is the largest square such that if unity be subtracted from each of its digits and again from each digit of the remainder, etc., all zeros being suppressed, each remainder is a square. Denote (10*—1)/(10 — 1) by {&}. Then ji(s+l)}2 has x digits and exceeds {x} by 10{|(x-l)}2. Since zeros are suppressed we have a square as remainder, and the process can be repeated. It is stated that therefore the property holds only for I2, II2, 1112, ....
Several1* found that 135 is the only number N composed of three digits in arithmetical progression such that the digits will be reversed if 132 times the middle digit be added to N.
W. Saint17 found the least integral square ending with the greatest number of equal digits. The possible final digits are 1, 4, 5, 6, 9. Any square is of the form 4n or 4n+l. Hence the final digit is 4. If the square terminated with more than three 4's, its quotient by 4 would be a square ending with two Ts, just proved to be impossible. Of the numbers ending with
'Arithmetic, both in Theory and Practice, ed. 4, London, 1727, 322.
10The Gentleman's Diary, or Math. Repository, London, 1767; Davis' ed., 2, 1814, 123.
^Jour. Nat. Phil. Chem. Arts (ed., Nicholson), London, 24, 1809, 124-6.
lcThe Gentleman's Diary, or Math. Repository, London, 1810, 38-9,'Quest. 952.
Wind., 1810, 39-40, Quset. 953.
lelbid., 1811, 33-4, Quest. 960.
l/Ladies' Diary, 1810-11, Quest., 1218; Leybourn's M. Quest. L. D., 4, 1817, 139-41.