Palindromic Smith Number/Examples/123,455,554,321/Mistake

Source Work

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary

$12,345,554,321$

Mistake

[$12,345,554,321$ is] a palindromic Smith number.

Correction

In fact it is not.

$12,345,554,321 = 41 \times 239 \times 271 \times 4649$

while:

$1 + 2 + 3 + 4 + 5 + 5 + 5 + 4 + 3 + 2 + 1 = 35$

but:

$4 + 1 + 2 + 3 + 9 + 2 + 7 + 1 + 4 + 6 + 4 + 9 = 52$

The mistake can be seen in the article by Underwood Dudley, where he states:

There we find palindromic Smith numbers, as $12345554321$, ...

In turn, he is quoting an article in Journal of Recreational Mathematics by Wayne L. McDaniel, which states:

We find that $R_2^2 = 121, R_5 R_8 = 123455554321, R_4 R_{10}, R_7 R_{12}, R_4 R_{20}, R_5 R_{24}, R_5 R_{31}$, and $R_4 R_{55}$ are palindromic Smith Numbers.

Indeed, $123 \, 455 \, 554 \, 321$ is a palindromic Smith number:

$1 + 2 + 3 + 4 + 5 + 5 + 5 + 5 + 4 + 3 + 2 + 1 = 40 = 1 + 1 + 4 + 1 + 7 + 3 + 1 + 0 + 1 + 1 + 3 + 7 + 2 + 7 + 1$

as:

$123 \, 455 \, 554 \, 321 = 11 \times 41 \times 73 \times 101 \times 137 \times 271$

Hence it appears that the transcription error was indeed made by Dudley.

Sources

• 1987: Wayne L. McDaniel: Palindromic Smith Numbers (J. Recr. Math. Vol. 19: pp. 34 – 37)

• Feb. 1994: Underwood Dudley: Smith Numbers (Math. Mag. Vol. 67, no. 1: pp. 62 – 65)  www.jstor.org/stable/2690561

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12,345,554,321$