# Infinite Sequence based on Simple Rule

Using one simple rule, an Infinite Sequence has been developed.

First 30 terms are given. Can you continue the sequence for at least next Ten Terms?

Series continues from top left to bottom right.

$$3, 31, 41, 59, 53, 89, 97, 79, 23, 43,$$

$$83, 79, 2, 41, 19, 97, 71, 37, 5, 97$$

$$59, 23, 7, 89, 3, 53, 11, 17 ,67, 79$$

$$\text{Next 10}$$?

• please change it to 30 terms May 26 '19 at 11:37
• Happy to see something different, this is a good start. May 26 '19 at 13:01
• Can you confirm that 51 is really correct and not a typo for, say, 53 or 61? May 26 '19 at 13:22
• Can you confirm if infinite sequence means an infinite sequence that does not repeat?
– tom
May 26 '19 at 14:05
• @tom..if you mean, cyclic..it is not..Sequence is not finite..of course numbers can repeat as you can see in the numbers given.
– Uvc
May 26 '19 at 14:23

The next terms are:

13, 23, 47, 5, 23, 31, 17, 53, 59, 11

Because

These are the primes found in pairs of digits of $$\pi$$.

31415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
|| | |   |  |||  |      |  |  |  |   ||||      |   |    |     | | |            |      |    |   ||  ||
3|41 |  53 89|| 23     43  | 79 02  41|71     37  05   97    59 |07           89     03   53  11| 67|
31  59      97|           83         19|                       23                              17  79
79                       97

• @florian..excellent!!
– Uvc
May 26 '19 at 15:35
• Thanks! It's the 3 numbers 31,41,59 that gave it away. After that it was a piece of pie. May 26 '19 at 15:39
• I could have made it tougher ...but I want to make it only slightly complex and a series not to be found in oeis...opens up for so many avenues of further research into this fascinating field
– Uvc
May 26 '19 at 15:44
• It would be nice to examine the prime gaps and frequency of various primes generated via this method through computer calculations.
– Uvc
May 27 '19 at 1:10
• The digits of pi being practically random, the frequency of each prime should be constant at 1 occurrence everu 100 digits. May 27 '19 at 12:44