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The guiding premise of this thread is the following proposition:

In other words, many seem to believe it impossible that we will ever be able to do more than simply estimate where the next prime may be found. I would contend that (although I certainly would not be the one to prove it..), in principle, it should be just as possible to locate the

Thoughts, as well as any related numerical observations, more than welcome...

- RF

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In relation to recent discussion here...

https://www.physicsforums.com/showthread.php?t=468910

https://www.physicsforums.com/showthread.php?t=467088

OBSERVATION

A SIMPLE ALGEBRAIC STATEMENT

= 5^(n+2) + (n + 2)^2 - (5 + (n + 2))

n = (0 --> 7)

for...

---------------------------------------------

5^2 + 2^2 - (5 + 2) =

---------------------------------------------

5^2 + 3^2 - (5 + 3) = 034 - 08 = 26

5^2 + 4^2 - (5 + 4) = 041 - 09 = 32

5^2 + 5^2 - (5 + 5) = 050 - 10 = 40

5^2 + 6^2 - (5 + 6) = 061 - 11 = 50

5^2 + 7^2 - (5 + 7) = 074 - 12 = 62

5^2 + 8^2 - (5 + 8) = 089 - 13 = 76

---------------------------------------------

5^2 + 9^2 - (5 + 9) =

---------------------------------------------

NOW COMPARE, keeping in mind this progression....

for...

----------------------------------

pi_(par_(

----------------------------------

pi (par_(13)) = pi (101) = 26

pi (par_(14)) = pi (135) = 32

pi (par_(15)) = pi (176) = 40 = 90 - pi_(par_(16)))

pi (par_(16)) = pi (231) = 50 = 90 - pi_(par_(15)))

pi (par_(17)) = pi (297) = 62

pi (par_(18)) = pi (

----------------------------------

pi_(par_(

----------------------------------

pi (par_(16)) - pi (par_(15)) = 50 - 40 = 10 --> 10 =

pi (par_(17)) - pi (par_(14)) = 62 - 32 = 30 --> 10 + 30 = 40 = 90 -

pi (par_(18)) - pi (par_(13)) = 76 - 26 = 50 --> 10 + 30 + 50 = 90 = 40 +

pi_(par_(

also...

pi_(par_(20)) = pi (627) = 114

= pi_(par_(

Are these relationships presented above (which, hopefully, I need not spell out...) "random" or "coincidental?" My reply: "Sure, they could be, but I find it rather unlikely." At the very least, it would be interesting to look at

**If fractals play a role in the behavior of partitions, then maybe, just maybe, they play a role also in the positioning of the primes**; and if they do, then who is to say that the two, prime numbers and partition numbers, cannot at some point down the road be mathematically related in precise manner via, for instance, the prime counting function and/or various number progressions related to fractals and/or the division of n-dimensional spaces?In other words, many seem to believe it impossible that we will ever be able to do more than simply estimate where the next prime may be found. I would contend that (although I certainly would not be the one to prove it..), in principle, it should be just as possible to locate the

*positions*of the primes as it is to calculate the*value*of the partition numbers.Thoughts, as well as any related numerical observations, more than welcome...

- RF

=================================================

In relation to recent discussion here...

**Ken Ono and Hausdorff dimensions**https://www.physicsforums.com/showthread.php?t=468910

and here...Hi All, In Ken Onos lecture he mentions Hausdorff dimensions appertaining to prime numbers:

5,7,11 relate to 0 dimension and primes from 13 to 31 as 1 dimension.

**relatively prime and independent confusion**https://www.physicsforums.com/showthread.php?t=467088

[tex]a^{10^{n+1}}-a^{10^n}\equiv\ 0\ mod\ 77[/tex]

[tex]a^{10^n}(a^{10^{n+1}-10^n}-1)\equiv\ 0\ mod\ 77[/tex]

[tex]a^{10^{n+1}-10^n}\equiv\ 1\ mod\ 77[/tex]

and [tex]10^{n+1}-10^n=10^n(10-1)=9\cdot\ 10^n[/tex]

since the above can be written as [tex]9\cdot\ 10^n=60k[/tex] for a natural k

[tex]a^{9\cdot\ 10^n}=a^{60k}\equiv\ 1\ mod\ 77[/tex]

by euler's theorem, if gcd(a,77)=1, since [tex]\varphi{(77)}=60[/tex] then

[tex]a^{60k}\equiv\ 1\ mod\ 77[/tex]

and it completes the proof

... I thought to pass along the following observation:Ken Ono cracks partition number mystery

https://www.physicsforums.com/showthread.php?t=465696

To avoid confusion, denote par_n as the n-th partition number. Then Ramanujan's congruences are...

par_(5k+4) == 0 (mod 5)

par_(7k+5) == 0 (mod 7)

par_(11k+6) == 0 (mod 11)

http://en.wikipedia.org/wiki/Ramanujan's_congruences

Note: 5, 7 & 11 are the 4th, 5th & 6th partition numbers and because they are all also prime (totient p) = (p-1), then their totient product: totient (5)*totient (7)*totient (11) = 240 = totient (5*7*11)

We can, in relation to your prior posting observe that...

par_(totient (77) + (1 + 77n)) =par_(60 + (1 +77n))

OBSERVATION

**The #31, The Golden Scale, The Prime Counting Function & Partition Numbers**A SIMPLE ALGEBRAIC STATEMENT

**20 + 2T_(n+1)**= 5^(n+2) + (n + 2)^2 - (5 + (n + 2))

n = (0 --> 7)

for...

*T_n denotes the n-th Triangular Number*---------------------------------------------

5^2 + 2^2 - (5 + 2) =

**029**- 07 = 22---------------------------------------------

5^2 + 3^2 - (5 + 3) = 034 - 08 = 26

5^2 + 4^2 - (5 + 4) = 041 - 09 = 32

5^2 + 5^2 - (5 + 5) = 050 - 10 = 40

5^2 + 6^2 - (5 + 6) = 061 - 11 = 50

5^2 + 7^2 - (5 + 7) = 074 - 12 = 62

5^2 + 8^2 - (5 + 8) = 089 - 13 = 76

---------------------------------------------

5^2 + 9^2 - (5 + 9) =

**106**- 14 = 92---------------------------------------------

**SUM = 400 = 20^2****106 - 29 = 77**NOW COMPARE, keeping in mind this progression....

**The Golden Scale (Fibonacci 2,5)****2, 5, 7, 12, 19, 31, 50, 81....**for...

*pi(n) - denotes the Prime Counting Function*

par_n - denotes the n-th Partition Numberpar_n - denotes the n-th Partition Number

----------------------------------

pi_(par_(

**12**)) = pi (**077**) = 21 =**(5 - 2) * 7**----------------------------------

pi (par_(13)) = pi (101) = 26

pi (par_(14)) = pi (135) = 32

pi (par_(15)) = pi (176) = 40 = 90 - pi_(par_(16)))

pi (par_(16)) = pi (231) = 50 = 90 - pi_(par_(15)))

pi (par_(17)) = pi (297) = 62

pi (par_(18)) = pi (

**385**) = 76----------------------------------

pi_(par_(

**19**)) = pi (490) = 93 =**(5 - 2) * 31**----------------------------------

**SUM = 400 = 20^2**pi (par_(16)) - pi (par_(15)) = 50 - 40 = 10 --> 10 =

**2 * 5**; 16 + 15 =**31**pi (par_(17)) - pi (par_(14)) = 62 - 32 = 30 --> 10 + 30 = 40 = 90 -

**50**; 17 + 14 =**31**pi (par_(18)) - pi (par_(13)) = 76 - 26 = 50 --> 10 + 30 + 50 = 90 = 40 +

**50**; 18 + 13 =**31**pi_(par_(

**19**)) - pi_(par_(**12**)) = 93 - 21 = 72 --> 10 + 30 + 50 + 72 = 162 =**2 * 81**; 19 + 12 =**31**also...

pi_(par_(20)) = pi (627) = 114

= pi_(par_(

**19**)) + pi_(par_(**12**)) = 93 + 26 = 114Are these relationships presented above (which, hopefully, I need not spell out...) "random" or "coincidental?" My reply: "Sure, they could be, but I find it rather unlikely." At the very least, it would be interesting to look at

*all*combinations of partition numbers, the index numbers of which sum to 5, 7, 11, 13 or 31, associated with Hausdorff dimension 0 and 1, just to see what relationships one might find. Some interesting patterns could emerge in relation to discrete intervals, just as there are interesting patterns that emerge in regards to prime gaps, etc...
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