The Number 137

The number 137 has raised considerable interest. Its reciprocal (1/137) approx. is referred to as the fine structure constant in physics and is related to the probability of electrons (or other particles) emitting or absorbing particles.

Much has been written regarding the "mystical" properties of this number. Indeed some years ago my attention was drawn to its significance through correspondence relating to Jungian archetypes.

And just recently an interesting article by Giorgio Piacenza has been published on Frank Visser's Integral World web-site.

Without wanting to claim too much for the "mystical significance" of this number, I would like to initially broaden the topic to highlight some important general properties of prime numbers (of which 137 is a specific example).

From one perspective prime numbers can be viewed as the basic building blocks of the natural number system (which we literally view in a linear manner as stretched out on a straight line).

However when we take the reciprocal of prime number a fascinating transformation is involved whereby a highly circular number pattern is involved, with the same sequence of digits continually recurring in a predictable manner.

So for example when we obtain the reciprocal of 7 the six digit sequence 142857 continually recurs.

This represents a fine example of a prime number whose reciprocal has a full periodic cycle.

A full cycle is measured as one less than the prime number in question which in this case gives a cycle of 7 - 1 = 6.

Though many other prime numbers e.g. 17, 61 and 97 exhibit full cycles in the digit sequence of their reciprocals, most prime reciprocals operate with a cycle of digits that is an exact fraction of the original prime number less one.

So when we look at the reciprocal of 137 we find that the eight digits 00729927 continually recur. So the cycle here represents 1/17 of 136. (i.e. 137 -1).

Indeed we can suggest a formula which universally holds for the sum of digits of these uniquely recurring sequences. If n = number of digits in sequence then the sum of digits = (9*n)/2.

Therefore in this case as n = 8, the sum of digits = 36.


What is particularly significant about this sequence is that ignoring the two zero digits we have in the remaining digits an example of a number palindrome.

Now a palindrome has an especially interesting psycho-mathematical connotation as a symbol of integration.

Thus whether we read the number sequence from the left or right direction, the number is the same. So these two polar opposites are thereby reconciled in a similar identity.

Some years ago I made a study of palindromes from this psycho-mathematical perspective and made a number of fascinating discoveries.

If we take any number and then obtain the difference with respect to its reverse (or mirror number) the result will always be divisible by 9.

Furthermore the resultant number will exhibit marked palindromic tendencies. Frequently the result will be an actual palindrome. In other cases adjustment of one or two digits by 1 will yield a palindrome!

To illustrate we will take 137. Now the reverse of this number (obtained by reading from right to left direction) is 731.
The difference is then 731 - 137 = 594.

When we divide this by 9 we obtain 66 which is a palindrome.

So a kind of numerical alchemy is at work here greatly facilitated through division by 9 which serves as a special catalyst of such integration.

Not surprisingly 9 forms the basis for what perhaps is the best known of modern personality systems i.e. The Enneagram (which places special emphasis on the manner through which the various personality types can achieve integration).

What then is special about 9?

Well, when we obtain the reciprocal we get the digit 1 continually recurring. So we have here in its reciprocal the simplest possible example of palindromic behaviour (for 1 is 1 whether read in a left hand or right hand direction).

If we confine ourselves to primes the next simple example (after 3) where its unique digit sequence reveals a palindrome is given by 11. So the reciprocal 09 reveals the single digit 9 (where the zero is ignored).

Indeed 11 likewise plays a unique integral role in number behaviour (which again holds universally).

Once again if we obtain the difference of any number and its mirror the result will always be divisible by 11 (where the initial number has an odd number of digits).

However where the initial number has an even count of digits, when we add both numbers the result will always be divisible by 11.

So again taking 137 (which is odd) the difference from its reverse i.e. 594 is indeed divisible by 11 = 54.

However if we take a number with an even count e.g. 2009 when we add this to its reverse we get 2009 + 9002 = 11011 which again is divisible by 11 = 1001.


Now a special significance of 137 is that - apart from the special case of 101 - it is the next prime number whose reciprocal reveals this unique palindromic digit sequence.

So 729927 is clearly a palindrome (with digit sequence the same when taken from left hand and right hand directions).

So when we divide this number equally into its left hand and right hand components, each component is the reverse of the other. So 729 (left hand) is the reverse of 927 (right hand) and vice versa. Also the sum of the first two digits (72) and last two (27) = 99 (the middle two digits).
It is also divisible by 99 with the result 7373.


Once again without wanting to attribute a direct physical significance to 1/137 (which is really only an approximation of the truer number 1/137.03604), it still possesses somewhat unique attributes.

So 137 as a prime number, represents a linear (independent) extreme. 1/137 represents a circular (interdependent) extreme possessing a palindromic sequence of several digits where L.H. and R.H. components are the reverse of each other.

It thereby possesses certain unique properties in psycho-mathematical terms as an archetype of integration.

Addendum 7/4/2026

Since writing this piece I have come to better realise the near magical properties of the reciprocals of prime numbers (in all number bases).

However sticking for the moment to the customary base 10, 00729927 (with its internal palindrome sequence) is the recurring 8 digit sequence of the reciprocal of 137 (i.e. 1/p where p = 137) Note here the square of the last two digits = the number representing the first 3 digits i.e. 27² = 729.

Now a fascinating property of all such even  sequences is that if we treat the unique repeating digits as comprising LHS and RHS numbers (with number of digits divided equally on both sides) and subtract the square of the LHS from the square of the RHS before then dividing by p – 2, we obtain the original number.

So in this case (9927² – 0072²)/135 = 00729927 (with the first two digits redundant in this case),

In fact this could be written in a form that uses only digits of the existing number,

i.e. (9927² – 0072²)/{27*(7 – 2)} = 00729927

 

Also if we subtract the LHS from the RHS once again dividing by p – 2, we always obtain a result that is just one more than the LHS number.

Thus, again in this case, (9927 – 0072)/135 = 73 i.e. (0073).

Interestingly 00729927/73 = 9999 just as 00729927/7373 = 99.

 

A simple equation, pn² – 2n governs all the numbers based on the unique recurring even digit sequences of reciprocals of the primes.

And when n = 73, 137n² – 2n = 00729927

So 00729927 (i.e. 729927) is remarkably the 73rd term in the sequence governed by 137n² – 2n.

So if we apply the same thinking to the best known cyclic number i.e. 142857 which is the unique digit sequence of 1/p where p = 7, then 142857 is the 143^rd term in the sequence governed by 7n² – 2n, so that when n = 143, 7n² – 2n = 142857.