I have done extensive research on cyclic numbers.I wish to present some facts/data.
1) Each cyclic number comes from taking the reciprocal of a prime like 7,13,17,31,41 ...etc or a reciprocal of a composite of primes like 91,49,133,343,1729...We can call these numbers as generators.Thus 7 is a generator of 142857,13 is a generator of 076923..etc
2) Generally for a generator like n=7,the cyclic number will have digits totalling to( n-1)=6.Thus for generator 17 you will have 16 digits,for 19 you will have 18 digits,for 97 you will have 96 digits,for 601 you will have 600 digits etc.
3) Alternately,and in much more number of cases,the cyclic number for 'n' will have digits totalling to a factor of
(n-1).Thus for 13,you will have 6 digits,for 31 you will have 15 digits,for 41 you will have 5 digits,for 73 you will have 8 digits etc
4) For a composite number like 287 which is a product of 7 and 41,you will have digits totalling to an LCM of 6 (relating to 7)and 5(relating to 41) or 30....
5) There are small numbers like 601 which have cyclic numbers of a large number (600) of digits,but there are also large numbers like 60249167 which have quite a lesser number of digits -to be exact 22 digits.-another example -1900381976777332243781 which has just 52 digits.
6) For some numbers like 127,you will have inter-relationship between the digits in the cyclic number.This has in fact 42 digits but you can split it up into 21 digits each-with the digits in the first group and those in the second group adding successively upto 9...Actually the cyclic number is 00787/40157/48031/49606/2 followed by 99212/59842/51968/50393/7....I have shown the digits seperated in groups of 5 to have a meaningful reading.
I will present some more interesting data in future blogs..
1) Each cyclic number comes from taking the reciprocal of a prime like 7,13,17,31,41 ...etc or a reciprocal of a composite of primes like 91,49,133,343,1729...We can call these numbers as generators.Thus 7 is a generator of 142857,13 is a generator of 076923..etc
2) Generally for a generator like n=7,the cyclic number will have digits totalling to( n-1)=6.Thus for generator 17 you will have 16 digits,for 19 you will have 18 digits,for 97 you will have 96 digits,for 601 you will have 600 digits etc.
3) Alternately,and in much more number of cases,the cyclic number for 'n' will have digits totalling to a factor of
(n-1).Thus for 13,you will have 6 digits,for 31 you will have 15 digits,for 41 you will have 5 digits,for 73 you will have 8 digits etc
4) For a composite number like 287 which is a product of 7 and 41,you will have digits totalling to an LCM of 6 (relating to 7)and 5(relating to 41) or 30....
5) There are small numbers like 601 which have cyclic numbers of a large number (600) of digits,but there are also large numbers like 60249167 which have quite a lesser number of digits -to be exact 22 digits.-another example -1900381976777332243781 which has just 52 digits.
6) For some numbers like 127,you will have inter-relationship between the digits in the cyclic number.This has in fact 42 digits but you can split it up into 21 digits each-with the digits in the first group and those in the second group adding successively upto 9...Actually the cyclic number is 00787/40157/48031/49606/2 followed by 99212/59842/51968/50393/7....I have shown the digits seperated in groups of 5 to have a meaningful reading.
I will present some more interesting data in future blogs..
Very interesting.
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