Problem of the Month (June 2001)

This month we feature a recreational number theory problem. The number 2592 is known as a printer error number. Why? Because if a printer tried to typeset 2592 and accidentally forgot to raise the exponents, there would be no error.

What are some other printer error numbers? What are the smallest printer errors in other bases? Are there any close misses? What if repeated exponentiation is allowed?

ANSWERS

Here is a list of all the known printer's errors in base 10 with no more than 20 digits. (A zero in parentheses indicates that the number remains a printer's error if any number of zeros are appended to the number)

Printer's Errors

Printer ErrorDigitsAuthor
25 924H. E. Dudeney
34 425(0)5D. L. Vanderpool
312 325(0)6D. L. Vanderpool
492 205(0)B. J. van der Zwaag
34 72 875(0)7B. J. van der Zwaag
10 74 4475(0)8B. J. van der Zwaag
13 74 5725(0)B. J. van der Zwaag
13 94 2125(0)B. J. van der Zwaag
16 74 6975(0)B. J. van der Zwaag
19 74 8225(0)B. J. van der Zwaag
1 73 7 72375(0)9E. Friedman
1 7 73 73875(0)E. Friedman
89 69 14944E. Friedman
90 69 92640E. Friedman
152 76 72 265(0)10E. Friedman
41850 97 875(0) E. Friedman
18103 176 75(0) M. Lapierre
1897 434 555(0) M. Lapierre
1757 574 1665(0) 11M. Lapierre
11745 78 20375(0)12E. Friedman
15348 78 26625(0)E. Friedman
18951 78 32875(0)E. Friedman
113 320 4930 325(0)13E. Friedman
13287635 712 96(0)E. Friedman
1653 7 77 286875(0)E. Friedman
185638 98 43125(0)E. Friedman
5 289 227976704014M. Lapierre
15072912 198 8875(0)15M. Lapierre
111 18 1211 331 1104617E. Friedman
10 1 916 1103868512 55(0) 18E. Friedman
14248618981 716 4287519M. Lapierre
1844674407370955 161620E. Friedman
30 720 22524580620 385(0)E. Friedman

And here is a list of some small printer's errors in other bases, due to Berend Jan van der Zwaag.

Printer's Errors in Other Bases

BasePrinter's Errors
211110 101(0),     1111 1000110,     111100 11110 1101(0)
32210 21 2(0),     1122 2211 22(0),     111 101 100110
4132 22(0),     121 2012,     132 132 31(0)
624,     112 53(0),     252 21(0)
7113 46 5(0)
833,     132 6(0),     114 70 6(0)
973 82 22(0)
1135 18(0)
1234 6(0),     252 5(0),     132 99(0)
1315 84 A(0)
1613 38 C(0)
1720 A4
1826 C(0),     53 D9(0),     1D2 A9(0)
2034 G(0),     74 14(0),     132 HA(0)

In 2017, Jean-Marc Falcoz generalized the notion of printer's errors to include horizontal as well as vertical shifts of digits, thus allowing additional multiplications. This gives the following additional solutions. Michael Yee added some more, including using repeated exponentiation.

Generalized Printer's Errors

35  7  21(0)
1  4  9  29  92
172  9  665(0)
17  6  9  47  2(0)
2  722  7  34
3  35  9  232
3 36  57  9  3(0)
34  72  875(0)
36  39  16  8(0)
67  184  6  4(0)
6  9  6  7  29  6(0)
7  58  73  28
10  74  4475(0)
10  75  64  80

Jean-Marc Falcoz even found this generalized pandigital printer's error: 35  1482  9760.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 2/23/17.