Magic in mathematics II

Author: Kasper B. Graversen, 28/07/09
Keywords: Fun
Abstract: Fun with the number cyclic numbers, and specifically with 142857 as it is the smallest of such numbers.
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Magic in mathematics II

A cyclic number which when multiplied with other numbers results in a cyclic permutation of the original number. In other words, the digits of a cyclic number always appear in the same order but will rotate around, when multiplied by a sequence of numbers. The smallest, and probably the most famous, of such numbers is 142857.

One of the aspects in math that I like, is that seemingly random and/or uninteresting numbers holds vast resources of fun. 142857 could be an old phone number (you know, back in the good old days, where a telephone was a rarity) or be some spare part to your car that you can't live without. However, the number also hold hidden properties. It is the smallest cyclic number. Let's see what fun it can bring us.

Multiplication fun with 142857

    142857 * 1 = 142857
    142857 * 2 = 285714
    142857 * 3 = 428571
    142857 * 4 = 571428
    142857 * 5 = 714285
    142857 * 6 = 857142
Notice how the number simply "shifts" one position to the left. The fun stops at 6 unfortunately as 
    142857 * 7 = 999999
The explanation for this mind boggling multiplication adventure lies in the fact that 1/7 = 0.142857 142857 142857...

Division fun with 7

We can observe similar behaviour when we divide various numbers by 7. Look at the below.. 
    1/7 = 0.142857 142857...
    2/7 = 0.285714 285714...
    3/7 = 0.428571 428571...
    4/7 = 0.571428 571428...
    5/7 = 0.714285 714285...
    6/7 = 0.857142 857142...

Being creative with 142857

People with more time for math than I have, have discovered various creative properties of 142857. I have taken the liberty to give these examples some funny titles. Bear with me.

"The reach of 9's" 
142857 * 7 = 999999
and
142 + 857 = 999 or 14 + 28 + 57 = 99

"8 is almost there..." 
142857 * 8 = 1142856
Removing the first digit and adding it to the last digit yields 142857

"15 can't cut it either..." 
142857 * 15 = 2142855
Removing the first digit and adding it to the last digit yields 142857

"Square and back again" 
142857 * 142857 = 20408122449
and
20408 + 122449 = 142857
142857 can also be found when performing even more permutations on the results. If you have the guts, try playing around with 142857 * 9 = 142857, 142857 * 11 = 1571427 or 142857 * 12 = 1714284 and see how 142857 can be found.

Cyclic numbers in general

There are many cyclic numbers. For example: 7 (shown above yielding the sequence 142857), 17 (yielding the sequence 0588235294117647), 19 (yielding the sequence 052631578947368421), 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, ... In fact there are infinitely many of them. I shall not bore you with the details of all of them or how you can find them. If you are mathematically inclined, you may find joy in visiting wikipedia on cyclic numbers or wikipedia on 142857 Have fun and please contact me if you have found more fun properties in 142857.

See also

For more math-related fun dig into Magic in mathematics I



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