The document describes tracking distances between digits of pi to find prime numbers. It alternates between digits of pi and e, recording distances and comparing them to differences between consecutive Riemann zeros. A set of deltas is built and analyzed for patterns related to prime number distribution up to the first few primes. Peaks in the deltas appear to correspond to prime numbers in order.

1. Foraging for Prime Number
amongst the digits of π
Subhendra Basu

2. ▪ 3141592653589793238462643383279502884197169399...
▪ 2718281828459045235360287471352662497757247093...
▪ We will be traversing the digits of π and e in an alternating
fashion... This traversal will be over the digits of π. We will
record the distance covered and take its difference with
consecutive Riemann Zeros as depicted in
: http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1

3. ▪ From the previous slide, the consecutive digits in the path
followed is:
▪ 3,7,4,8,5,8,2,8,5,8,5,5,9,0,9,5,2,3,8,3,6,0,6,8,3,4,8,1,2,5,9,6,0,
2,8,9,4,7,9,7,1,4,9,0,9,3,….

4. ▪ {31415926535897}
▪ len("31415926535897") = 14
▪ Riemann Zero Counter = 1
▪ Riemann Zero = 14 [Integral Part]
▪ Delta['14'] = 0

5. ▪ len("31415926535897932384") = 20
▪ R[2] = 21
▪ Delta["21"] = -1
▪ Set of delta(s) = { 0, -1, …}

6. ▪ len("314159265358979323846264338") = 27
▪ Delta['25'] = +2
▪ DS = { 0, -1, +2,…}

7. ▪ len("31415926535897932384626433832795")
▪ Delta["30"] = -2
DS = { 0, -1, +2, -2, … }

8. ▪ len("314159265358979323846264338327950288") = 36
▪ Delta["32"] = +4
▪ DS = { 0, -1, +2, -2, +4,… }

9. ▪ len("31415926535897932384626433832795028841971693
99375") = 49
▪ Delta["37"] = 12
▪ DS = { 0, -1, +2, -2, +4,12, … }

10. ▪ len("3.141592653589793238462643383279502884197169
399375105") = 53
▪ Delta["40"] = 13
▪ DS = { 0, -1, +2, -2, +4, 12, 13… }

11. ▪ len("31415926535897932384626433832795028841971693
993751058209") = 56
▪ Delta["43"] = 13
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13,… }

12. ▪ len("31415926535897932384626433832795028841971693
9937510582097494459230") = 66
▪ Delta["48"] = 18
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18,… }

13. ▪ len("31415926535897932384626433832795028841971693
9937510582097494459230781640628620899") = 81
▪ Delta["49"] = 32 [Riemann Zero]
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32… }
▪ Lookup DS @5th index, "32" = 4 [saved as a bookkeeping
exercise] ,

14. ▪ len("31415926535897932384626433832795028841971693
99375105820974944592307816406286208998628034825"
) = 91
▪ Delta["52"] = 39
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39… }

15. ▪ len("31415926535897932384626433832795028841971693
99375105820974944592307816406286208998628034825
342") = 94
▪ Delta("56") = 38
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38… }

16. ▪ len(31415926535897932384626433832795028841971693
99375105820974944592307816406286208998628034825
342117067982148086513") = 112
▪ Delta("59") = 53
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53…
}

17. ▪ len("31415926535897932384626433832795028841971693
99375105820974944592307816406286208998628034825
34211706798214808651328") = 114
▪ Delta("60") = 54
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, … }

18. ▪ len("31415926535897932384626433832795028841971693
99375105820974944592307816406286208998628034825
3421170679821480865132823") = 116
▪ Delta("65") = 51
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, 51, … }

19. ▪ len("31415926535897932384626433832795028841971693
99375105820974944592307816406286208998628034825
3421170679821480865132823066") = 119
▪ Delta("67") = 52
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, 51, … }
▪ DS Lookup, index = 11; is equal to 39, saved
▪ So far, {4, 39,…}

20. ▪ len("31415926535897932384626433832795028841971693
99375105820974944592307816406286208998628034825
3421170679821480865132823066470") = 122
▪ Delta("69") = 53
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, 51, 52, 53 … }

21. ▪ len("31415926535897932384626433832795028841971693
99375105820974944592307816406286208998628034825
3421170679821480865132823066470938446") = 128
▪ Delta("72") = 56 (Riemann Zero)
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, 51, 52, 53, 56 … }

22. ▪ len("31415926535897932384626433832795028841971693
99375105820974944592307816406286208998628034825
34211706798214808651328230664709384460955058") =
135
▪ Delta["75"] = 60, Riemann Zero
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, 51, 52, 53, 56, 60 … }
▪ DS [lookup, index 14] = 54

23. ▪ len("31415926535897932384626433832795028841971693
99375105820974944592307816406286208998628034825
34211706798214808651328230664709384460955058223"
) = 138
▪ Delta("77") = 61
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, 51, 52, 53, 56, 60, 61 … }

24. ▪ len("31415926535897932384626433832795028841971693
99375105820974944592307816406286208998628034825
34211706798214808651328230664709384460955058223
17253594") = 146
▪ Delta("79") = 67, riemann zero
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, 51, 52, 53, 56, 60, 61, 67 … }

25. ▪ @Slide 14,
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, … } ==> This gives the distribution of Prime numbers
upto first 4 primes.
▪ @ Slide 21, what do you make out of it before we go to next?
▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, 51, 52, 53, 56, 60, 61, 67 … }

26. ▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, 51, 52, 53, 56, 60, 61, 67 … }
▪ The pattern that unfolds is: [Number of Primes to be
decoded] followed by set of primes , I mean their
distribution
▪ Green color indicates hits where differences are Zeros
▪ Red color indicates lookup values/Primes in a symbolic
fashion

27. ▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, 51, 52, 53, 56, 60, 61, 67 … }
▪ Yellow color indicates both hits and lookup for other hits.
▪ So where are the primes and distribution of prime numbers?
▪ +4 = number of primes to be decoded in this run
▪ The subset starting 32, ending 52, look closely at it.

28. ▪ DS = { 0, -1, +2, -2, +4, +12, +13, +13, +18, +32, 39, 38, 53,
54, 51, 52, 53, 56, 60, 61, 67 … }
▪ Subset of DS={+32, 39, 38, 53, 54, 51, 52};
▪ 32 is in green signifying start
▪ 52 is in yellow signifying stop

29. ▪ Prime Number Distribution:
▪ 1-to-1 correspondence:
▪ +32, 39, 38, 53, 54, 51, 52
▪ 1, 2, 3, 4, 5, 6, 7
▪ C, P, P, C, P, C, P

30. Any Feedback is welcome:
basu.subhendra@outlook.com