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142857 2 = 142857 × 142857 = 20408122449 20408 + 122449 = 142857. Multiplying by a multiple of 7 will result in 999999 through this process: 142857 × 7 4 = 342999657 342 + 999657 = 999999. If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number. [citation ...
If leading zeros are not permitted on numerals, then 142857 is the only cyclic number in decimal, due to the necessary structure given in the next section. Allowing leading zeros, the sequence of cyclic numbers begins: (10 6 − 1) / 7 = 142857 (6 digits) (10 16 − 1) / 17 = 0588235294117647 (16 digits)
For any integer coprime to 10, its reciprocal is a repeating decimal without any non-recurring digits. E.g. 1 ⁄ 143 = 0. 006993 006993 006993.... While the expression of a single series with vinculum on top is adequate, the intention of the above expression is to show that the six cyclic permutations of 006993 can be obtained from this repeating decimal if we select six consecutive digits ...
The vinculum, in its general use, was introduced by Frans van Schooten in 1646 as he edited the works of François Viète (who had himself not used this notation). However, earlier versions, such as using an underline as Chuquet did in 1484, or in limited form as Descartes did in 1637, using it only in relation to the radical sign, were common.
3 Found number PI (3,1415) in 142857. 4 What will be the Factorial of 285714 (i.e. 285784!) 2 comments. 5 Very interesting number! 1 comment. 6 Other properties. 1 ...
If one knows that π is approximately 3.14159, then it trivially follows that π < 22 / 7 , which is approximately 3.142857. But it takes much less work to show that π < 22 / 7 by the method used in this proof than to show that π is approximately 3.14159.
Some Egyptologists [3] have claimed that the ancient Egyptians used an approximation of π as 22 ⁄ 7 = 3.142857 (about 0.04% too high) from as early as the Old Kingdom (c. 2700–2200 BC). [4] This claim has been met with skepticism.
107 is the 28th prime number.The next prime is 109, with which it comprises a twin prime, making 107 a Chen prime. [1]Plugged into the expression , 107 yields ...