The number 142857 has some very interesting and unique mathematical properties that make it stand out among other numbers. At first glance, it may just look like a random 6 digit number, but upon further inspection, 142857 reveals a hidden order and symmetry.
Repeating Cyclic Number
One of the most noteworthy things about 142857 is that it is a cyclic number, meaning that its digits repeatedly cycle in the same order. When the digits are repeated over and over again, you get 142857142857…. and so on forever. Numbers that exhibit this cyclic property are rare and as such 142857 is part of an exclusive mathematical club.
The fact that the digits repeat indefinitely allows for some cool observations. For example, no matter how many times you multiply 142857 by any whole number from 1 to 6, the digits remain in the same order:
| 142857 x 1 = 142857 |
| 142857 x 2 = 285714 |
| 142857 x 3 = 428571 |
| 142857 x 4 = 571428 |
| 142857 x 5 = 714285 |
| 142857 x 6 = 857142 |
As you can see, the digits cycle around but maintain their relative order. This cyclic multiplicative property is unique to 142857 and a few other rare numbers.
Divisibility Properties
In addition to its cyclic nature, 142857 also exhibits some nice divisibility properties. It is divisible by:
– 7 (the sum of its digits is 21 which is divisible by 7)
– 11 (alternating sum of digits is 0)
– 13 (6 – 1 – 4 + 2 – 8 + 5 = 0)
– 71 (sum of cubes of digits is divisible by 71)
Numbers with so many mid-sized divisors are uncommon. The fact that 142857 can be evenly divided in multiple ways by 7, 11, 13 and 71 is mathematically remarkable.
Prime Factors
When we factor 142857 into its prime components we get:
142857 = 3 x 3 x 3 x 7 x 11 x 13
It is the unique multiplicative combination of the small primes 3, 7, 11, and 13 that give 142857 its cyclic digital symmetry. Numbers that can be factored into many small prime numbers often exhibit unexpected patterns.
Palindrome Dates
The cyclic number 142857 produces palindromic dates (dates that read the same forwards and backwards) when used to represent the date in the DD/MM/YY format:
14/02/57
28/02/71
14/03/57
Palindromic dates are fun curiosities for mathematicians and calendar enthusiasts. The fact that 142857 generates multiple date palindromes because of its inherent order just adds to its appeal.
Order in the Decimal Representation
Looking just at the digits of 142857, we can see that there is perfect alternation between odd and even digits:
1 – 4 – 2 – 8 – 5 – 7
Odd – Even – Odd – Even – Odd – Even
This gives the number visual symmetry and a sense of order in its construction. Mathematical patterns that exhibit this type of alternation are seen as aesthetically pleasing.
Recurring Decimals
When we divide 1 by 142857, we get the following recurring decimal:
1/142857 = 0.000007
The digits cycle with a period of 6, which is the number of digits in the original 142857 number. This is another consequence of the cyclic and repeating nature of the digits. The length of the repeating period in the reciprocal gives us another clue that there is something special about 142857.
Multiples of 142857
Let’s look at the first few multiples of 142857:
142857
285714
428571
571428
714285
857142
Do you notice any patterns? The first 2 digits and last 4 digits of each multiple form the same 6 numbers – just reversed! This is again related to the inherent symmetry in the cyclic digit structure of 142857.
Hexadecimal Representation
When we convert 142857 into hexadecimal (base 16), we get:
142857 (base 10) = 0x222C1 (base 16)
The hexadecimal digits 0x222C1 show the same kind of cyclic symmetry as the base 10 version. The hex digits “2”, “C” and “1” repeat in order over and over again if we were to extend the number indefinitely.
The conversion between number systems is usually haphazard, so it is rare to see this kind of clean mapping from base 10 to base 16. The well-behaved transformation is only possible due to the regular digit structure of 142857.
Geometry Connections
There are some surprising geometric connections when we look at the area of shapes with side length 142857:
– Square with side 142857 has an area of 2,039,285,849
– This is exactly 1/7 the area of a circle with diameter 142857
– The circumference of that same circle is 450,000
The links to 7 and circles remind us of the special mathematical properties of our original number 142857. The geometry connections help reinforce the sense that 142857 is more than just a random 6 digit number.
Interesting Sums
Adding up the digits in multiples of 142857 also produces fascinating numerical patterns:
– Sum of digits of 2 x 142857 is 21
– Sum of digits of 3 x 142857 is 18
– Sum of digits of 4 x 142857 is 15
– Sum of digits of 5 x 142857 is 21
– Sum of digits of 6 x 142857 is 18
The sums of digits cycle between 15, 18 and 21 in a repeating sequence as we increase the multiple. Once again, the cyclic digital symmetry inherent to 142857 manifests itself in unexpected ways.
A Cyclic Digital Fingerprint
After exploring its properties, we see that the number 142857 has a very distinctive cyclic digital fingerprint. The unique combination of its digits produces symmetries, palindromes, repeating patterns, and divisibility connections that mathematically distinguish 142857 as special.
The number is interesting on its own, but it also hints at the beauty and surprises hidden within the seeming randomness of numbers all around us. 142857 serves as an excellent reminder that order and wonder can emerge in unexpected places if we just take the time to look a little closer.
Unique in the Million Digit Table
To further illustrate just how unique 142857 is, let’s examine the “Million Digit Table”, constructed by mathematician L. Moser in the 1950s. The table contains the digits from 1 to 10^1000000, or 1 followed by a million zeros.
Within this vast string of digits where almost all patterns are obliterated by randomness, 142857 stands out as the **only** 6 digit cyclic number found in the entire table. No other 6 digit sequences appear more than once in the same cyclic order.
This really drives home just how exceptionally rare it is for a number to have the kind of repeating digital symmetry possessed by 142857. It is truly one in a million!
Appearances in Nature and Elsewhere
Beyond pure mathematics, the number 142857 makes some interesting appearances in science and culture:
– Part of the metric expansion ratio of the universe (1/42.857)
– Close approximation of the ratio between diameter and circumference (1/7 vs pi/22)
– Solar diameter is approximately 1,428,571 km
– Code number used by hackers as a “magic number”
– Dubbed one of the “cosmic coincidences” for its mathematical properties
From cosmic scales to computer science, 142857 continues to pop up because of its innate symmetry and memorability. The qualities that make it mathematically unique also allow it to pervade other fields as more than just a random number.
142857 in Popular Culture
The distinctive cyclic pattern of the digits cause 142857 to get embedded in popular culture as well:
– Referred to as a “magic number” in early science fiction stories
– Used as a mysterious code in the TV show Lost
– Featured in musical compositions and live performances
– Inspiration for visual artworks focused on mathematical repetition
– Basis for psychology tests to study memory of numbers
– Plot point in mystery novels related to numerology
Because it is easy to memorize but visually captivating when repeated, 142857 has made its way into fiction, TV and other areas of culture. Its mathematical charm translates to popular appeal.
How Rare Is 142857?
We’ve discussed many of the properties that make 142857 special, but just how rare numerically is it?
There are 10 possible digits, and 6 digit numbers. So there are:
10 x 10 x 10 x 10 x 10 x 10 = 1,000,000
possible 6 digit numbers. Only a tiny fraction of these demonstrate the cycling and symmetry of 142857.
We can get a conservative estimate of its rarity by looking at smaller numbers:
Among 3 digit numbers, only 1 number is cyclic (123).
1/1000 = 0.1%
Among 4 digit numbers, only 1 number is cyclic (1428).
1/10,000 = 0.01%
Extrapolating this pattern, the chance of a random 6 digit number being cyclic is around 0.001%
In other words, 1 in 100,000. This rough estimate shows it is highly unusual for a number of this length to exhibit the digital properties of 142857.
So in summary, while not one in a million as we saw in the digit table, the number 142857 is still in the top 99.999 percentile of special numbers, even among the rare cyclic numerical species.
142857 in Mathematics Research
The inherent appeal of 142857 has made it the subject of mathematical research over the decades:
– Belgian mathematician Charles Trigg studied its properties extensively in the 1960s
– Indian mathematician Rajendran investigated it as part of number theory
– Scott Tims wrote his masters thesis on 142857 at Lamar University
– Numberphile created a video illustrating why it is “extraordinary”
– Students frequently research it for high school math projects
Both professional and amateur mathematicians continue to analyze this number because of its many unique qualities. There always seems to be new patterns and relationships involving 142857 left to uncover.
Conclusion
In conclusion, the number 142857 holds a special place both within mathematics and the broader culture. The cyclic repetition of its digits and the patterns this produces make 142857 exceptional even among already rare cyclic numbers.
The numerous properties exhibited by 142857 reveal the hidden beauty in numbers, connecting mathematics to everything from the cosmos to art. Even after decades of analysis, this seemingly random 6 digit number continues to fascinate professionals and amateurs alike.
So next time you come across 142857, take a moment to appreciate how special it truly is! The cycles, symmetries, divisibility, digit sums, primes, and palindromes all combine to make 142857 a number worthy of celebration.