Cowdery Numbers


I have discovered (during my A level maths classes instead of listening to the teacher!) a series of number pairs, the reciprocals of which contain the same digits after the decimal point as the other number in the pair. I discovered these by firstly noting that sqrt(2)+1 had the same bits after the decimal point as 1/(sqrt(2)+1). Random calculator fiddling showed that this property wasn't a unique result.

For example,

And similarly:



A general formula for generating the numbers is as follows:

Where n is any integer (or an integer + 0.5 - see proof B).

Proof A

- Proof that the reciprocal of one number is the other

Equate the two numbers

Cross multiply

Multiply out



Proof B

- Proof that the two numbers are identical after the decimal point for an infinite number of decimal places

To prove this, simply show that the difference between the two numbers is an integer. This can be shown by subtracting the two numbers:

The difference between the two numbers is 2n, and since n is an integer, so is 2n, and thus the two numbers in the Cowdery pair are indeed identical to an infinite number of decimal places. It follows that as 2n must be an integer, n can be an integer + 0.5 and the theory holds.

The real proof

The real proof is of course:

What is the set of C for which k is an integer.

Rearranging into standard quadratic form:

Which can be solved using the standard formula for quadratics where a=1, b=-k and c=-1. This gives C in terms of k:

Which is the same as the equations shown above.

And this gives the results shown in the table below:


Here are the first 21 Cowdery Number pairs:

Cowdery Number (n)< 1> 1
0.5 0.6180339889 1.6180339889
1 0.4142135624 2.4142135624
1.5 0.3027756377 3.3027756377
2 0.2360679777 4.2360679777
2.5 0.1925824033 5.1925824033
3 0.1622776603 6.1622776603
3.5 0.1400549449 7.1400549449
4 0.1231056247 8.1231056247
4.5 0.1097722296 9.1097722296
5 0.0990195144 10.0990195144
5.5 0.09016994387 11.09016994387
6 0.08276253007 12.08276253007
6.5 0.07647321932 13.07647321932
7 0.07106781192 14.07106781192
7.5 0.06637297571 15.06637297571
8 0.0622577481 16.0622577481
8.5 0.0586213842 17.0586213842
9 0.05538513884 18.05538513884
9.5 0.05248658732 19.05248658732
10 0.04987562075 20.04987562075

Of course the keen eyed amongst you will have observed that the Cowdery number for n=0.5 is the Golden Ratio, known to the ancient Greeks.


Are these numbers already known about?

Do they have any use?

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Last modified on: 25th August 2017 by email the webmaster

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