## James Booth's 2 sets of 3 digit numbers

*abcabc*= 1001.

*abc*

Since 1001 = 7.11.13 the result is clear.

It isn't clear what he means by this. If 2Like properties may be found for17,19,23, but the periods are longer. The prime divisor being2n +1, it is manifest the number of places in the period cannot exceed; however it may fall short of n. Thus when the divisor is17, the number of places in the period is eight.

*n*+ 1 only refers to 17, 19, 23 then he is correct. The periods are 8, 9 and 11. However, these are exactly

*n*and they do not fall short. Clearly 2 sets of

*n*digits gives numbers divisible by the factors of 10

^{n}+ 1 so these facts are easy to check. If, however, Booth means that the result hold for all other values of

*n*where 2

*n*+ 1 is prime then he is incorrect. For example, it is false for 2

*n*+ 1 = 31.

Now if *p* is an odd prime, say *p* = 2*n* + 1, then 2 sets of *n* digits is divisible by *p* for *p* = 7, 11, 17, 19, 23, 29 but not for 31, 37, 41, 43. For 13 we get periods of 3 + 6*k* but not 6. For primes 47, 59, 61 again we get period (*p* - 1)/2 but not for 53, 67, 71. Again we get period (*p* - 1)/2 for 73, 97, 101, 103, 109, 113 but not 79, 83, 107. For 89 we get a period of 22 but not of 44.

JOC/EFR February 2016

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