I stumbled upon such a number (no pun intended) in an article in The Times of India dated 16th June 2025, that provides the same cheer to mathematicians and statisticians alike. Rumoured to be the favourite number of magicians, and even associated with a million dollar maths mystery. Spanish myth calls it ‘Número Cabalístico’—a number that has mystical or occult significance in Kabbalah or Judaism. The numerical value assigned to Hebrew alphabet (Gematria) is said to reveal hidden spiritual meaning and energies of the person. This number was unearthed by archaeologist in the many Egyptian pyramids built around 25th century BCE.
“Sometimes, the most profound tools are hidden in plain numbers. It invites learners into playful discovery. It shows modular arithmetic in action,” states Manjul Bhargava, a renowned mathematician and Fields Medal* winner.
Curiouser, Shakuntala Devi—the world’s fastest human computer—loved this number. She used it predictably to read astrology charts and found patterns in the poetry of this number. Calling it “Ghumakkad Ank”—the revolving number—was her ode to the rhythmic divine digit 142857. The next such magical number is a 16-digit one. Hidden in plain sight, this cyclic number boggles your mind with its easter eggs to find and discover amongst its fold.
So, what is a cyclic number?
A cyclic number is an integer where its multiples, when multiplied by 1, 2, 3, …, n (where n is the number of digits in the number), produce the same digits in a cyclic order.
Key characteristics of cyclic numbers:
- Cyclic permutation of digits: As you multiply the number by integers up to its number of digits, the digits in the product are just a cyclic rearrangement of the original number’s digits.
- Connection to repeating decimals: Cyclic numbers are often generated from the repeating part of the decimal expansion of the reciprocal of a prime number. The smallest prime number is 7. The reciprocal therefore will be 1/7 = 0.142857142857… generates the cyclic number 142857. Multiplying this number by integers from 1 to 6 results in cyclic permutations of its digits.
- Row of 99’s: When a cyclic number is multiplied by its generating prime, the product is always a row of 9s. Exactly nine primes smaller than 100 generate cyclic numbers: 7, 17, 19, 23, 29, 47, 59, 61, 97.
We can search for cyclic numbers by dividing 1 by various prime numbers and looking at the pattern of repeating decimals that we obtain. So, are there infinite cyclic numbers? This, remarkably, is an unsolved problem.
The Illustrious 142857
When multiplied by 1 through 6, the digits of 142857 are rearranged in a circular fashion:
142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142
Diving Deeper
Divide 1 by 7:
1 ÷ 7 = 0.142857142857…
The digits 142857 repeat endlessly in the exact same order, making us look for patterns of symmetry and symphony in the mundane maths. Now let’s try 2, 3, 4, 5, and 6:
- 2 ÷ 7 = 0.28571428571…
- 3 ÷ 7 = 0.42857142857…
- 4 ÷ 7 = 0.57142857142…
- 5 ÷ 7 = 0.71428571428…
- 6 ÷ 7 = 0.85714285714…
The design that emerges is 1/7 = 0.142857 recurring and has order 6 (i.e., is 6 digits long). It follows that 1000000/7 = 142857.142857 recurring = 142857 + 1/7. Multiplying by 7 we find that 142857 × 7 = 999999.
Furthermore, when 142857 is multiplied by itself, the result is 20408122449. Split into two parts, 20408 and 122449, and when added (20408 + 122449), we come back to 142857.
The next time when you claim a share of the pie in the pizza, notice how 142857 serendipitously peeks its head in π too!
22/7 = 3.1428571428…