Prime Reciprocals

[John G]

http://primereciprocals.net.au

AN EXPLORATORY STUDY OF THE RECIPROCALS OF SOME OF THE PRIMES
by Geoffrey Arthur Tingate 2007

SUMMARY
Links are established between pairs and trios of primes by treating their reciprocals as whole numbers and breaking these down into their primes. It is also shown that pairs and trios of primes are linked through 2 families of numbers, 111, 1,111, 11,111 ... and 101, 1,001, 10,001 ... .

The first primes to be so linked are 7 and 13, but larger linked primes are more disparate, ranging from the pair 73 and 137 to the trio 29, 281 and 121,499,449. The indications are that links of this kind are to be found beyond the limits of this study, and perhaps indefinitely.

 

[Rockimedes]

Thanks John, awesome site. I'm always interested in finding connections between the smaller integers with larger ones. A fractal-type of arithmetic where one can deduce properties of huge numbers by analyzing our regular integers would be my dream come true.

Another test for divisibility by 11 that I'm surprised he didn't mention is this:
take a number in decimal form n=ar....a3a2a1a0 Then for the sum m=a0-a1+a2-a3+...+ar. If 11 divides m, then 11 divides n.
Example: n=7197432 then m=2-3+4-7+9-1+7=11, so 11 does divide 7197432.

This all only works since we use base-10. If we used something else like base-12 or 60 there would be a lot more divisibility rules. It would be interesting to see what would happen if he tried different bases in his studies...looks like I have found something to keep myself busy with for the summer.