Infinite Semi-Pronic Prime Series Appears to Converge

[Rudy Toody]

A while back I posted a question about this series on the General Math forum and was brought to task for not showing any math.

My hope is to prove that these series are infinite.

http://oeis.org/A002378" are the series 0,2,6,12,20,30... and distances between consecutive numbers are increasing even spans. The mid-points of those spans are consecutive squares.

This is the main series where we combine the low half-pronic span with the high half-pronic span for each step.
[PLAIN]http://math.rudytoody.us/LogVersionX.png
x is: 1.500989... which equals y + z

This is a convergent series that involves only the high half-pronic spans.
[PLAIN]http://math.rudytoody.us/LogVersionY.png
y is: 0.851916...

This is a convergent series that involves only the low half-pronic spans.
[PLAIN]http://math.rudytoody.us/LogVersionZ.png
z is: 0.649073...

It is obvious that if a span was prime-free, these sums would be greater than 1 and the series would diverge.

This is what I call the Bertrand Series, based on Bertrand's Postulate, which states that n < p <= 2n and has been proved several times. I use 2^n < p <= 2^n+1 to create the series.
[PLAIN]http://math.rudytoody.us/LogVersionW.png
w is: 0.868897...

Because it it known that there are one or more primes between each power of two, no exception can cause the series to diverge.

Plotting the first dozen steps of each series shows a line up for 2 steps and then an abrupt right turn into a horizontal line. This is what makes me think they converge.

Because each step produces a square-free denominator whose factors are the primes in that span, every step will be co-prime to all other steps and to the denominator of the sum. The numbers can be summed in any order. Because the denominators are made up of the product of the primes for each span, occasionally step n will be less than step n+1. But, the overall trend is decreasing.

I have validated each sum to over 500,000 unchanging digits after about 10-20 steps.

My questions are:

1) What is required to prove convergence?
2) What would it take to prove that a prime-free span cannot occur for the pronic series?

 

[Rudy Toody]

This series, if proved, will lead to proofs of a few long-standing conjectures.

You can do magical things with this series.

If the reader has an AHA! moment and finds the proof, he or she is welcome to publish it.

I would appreciate this citation:

Kline, Fred Daniel "Semi-Pronic Prime Series Problem" Private Correspondence, http://math.rudytoody.us

 

[Rudy Toody]

I have discovered that the series I found is actually Opperman's Conjecture. (1882) And it is stated that if it is proved, the others I mentioned will also be proved.

So, if we can prove that the series is infinite because an exception (missing prime within a gap) will never occur, then we got it.

Any ideas?

 

[Raphie]

Rudy Toody,

I can't help you with your proof, but just wanted to say it's nice to see someone else looking at Pronic Numbers in relation to the primes. My interest in the Pronic Numbers stems from what I view as some rather unlikely "coincidences" in relation to this class of number and Kissing Numbers (K_n) / Mersenne Primes. Below is an example of what I mean:

1! * 2 = 002 = 2*T_01 = K_1 --> 2*T_01 - 6*00 = 2*T_((K_1)/2) - 6*(0^2)
2! * 3 = 006 = 2*T_02 = K_2 --> 2*T_03 - 6*01 = 2*T_((K_2)/2) - 6*(0^2 + 1^2)
3! * 2 = 012 = 2*T_03 = K_3 --> 2*T_06 - 6*05 = 2*T_((K_3)/2) - 6*(0^2 + 1^2 + 2^2)
4! * 3 = 072 = 2*T_08 = K_6 --> 2*T_12 - 6*14 = 2*T_((K_4)/2) - 6*(0^2 + 1^2 + 2^2 + 3^2)
5! * 2 = 240 = 2*T_15 = K_8 --> 2*T_20 - 6*30 = 2*T_((K_5)/2) - 6*(0^2 + 1^2 + 2^2 + 3^2 + 4^2)

Also...
2*T_16 = 272 = K_9 (Lattice)
2*T_17 = 306 = K_9 (Non-Lattice)

A) 2,6,12,72,240 are all Highly Composite (HCN_n) Maximal known Kissing Numbers as well as maximal proven lattice Kissing Numbers
B) 2,6,12,72,240 are all the average of twin "qrimes" Definition: "qrime number": n in N | d(n) < 3. Notation: p'_n (In other words, 0 & 1 Union the Prime Numbers...)

for par_n denotes nth partition number

2*TRIANGULAR INDEX NUMBER + 1
2*01 + 1 = 03 = p'_2 --> 2nd Mersenne Prime Exponent
;2 = p'_(1-1) + 1 = par_1 + 1 = K_1/1 = d(2) = HCN_1

2*02 + 1 = 05 = p'_3 --> 3rd Mersenne Prime Exponent
;3 = p'_(2-1) + 1 = par_2 + 1 = K_2/2 = d(4) = HCN_2

2*03 + 1 = 07 = p'_4 --> 4th Mersenne Prime Exponent
;4 = p'_(3-1) + 1 = par_3 + 1 = K_3/3 = d(6) = HCN_3

2*08 + 1 = 17 = p'_7 --> 6th Mersenne Prime Exponent
;6 = p'_(4-1) + 1 = par_4 + 1 = K_4/4 = d(12) = HCN_4

2*15 + 1 = 31 = p'_11 --> 8th Mersenne Prime Exponent
;8 = p'_(5-1) + 1 = par_5 + 1 = K_5/5 = d(24) = HCN_5

2*DIMENSION NUMBER + 1
2*01 + 1 = 03 = p'_2 --> 2nd Mersenne Prime Exponent; 2 = (1+1)
2*02 + 1 = 05 = p'_3 --> 3rd Mersenne Prime Exponent; 3 = (2+1)
2*03 + 1 = 07 = p'_4 --> 4th Mersenne Prime Exponent; 4 = (3+1)
2*06 + 1 = 13 = p'_6 --> 5th Mersenne Prime Exponent; 5 = (4+1)
2*08 + 1 = 17 = p'_7 --> 6th Mersenne Prime Exponent; 6 = (5+1)
-------------------------------------------------------------------------------------
2*09 + 1 = 19 = p'_8 --> 7th Mersenne Prime Exponent; 6 = (6+1)

In short, I believe the "magic" of the Pronics extends well beyond their relationship to the prime number distribution. For instance, you will find with a bit of research that the Pronics are also related to the coordination sequences of lattices...

e.g.

A002378 Oblong (or promic, pronic, or heteromecic) numbers: n(n+1).
a(n) is the number of minimal vectors in the root lattice A_n (see Conway and Sloane, p. 109).
http://oeis.org/A002378Raphie

 

[blue_raver22]

I would approach this question by first examining the evidence presented and then considering the mathematical principles that could potentially support or refute the claim that the series converges.

Based on the information provided, there does appear to be a pattern in the series that suggests convergence. The fact that the distances between consecutive numbers are increasing even spans and the mid-points of those spans are consecutive squares is a strong indication that the series may converge. Additionally, the fact that the sums of the series are decreasing and have been validated to a high number of digits also supports the idea of convergence.

To prove convergence, we would need to show that the series approaches a finite limit as the number of terms increases. This can be done through various mathematical methods such as the limit comparison test or the Cauchy convergence criterion. It would also be important to consider the properties of the series, such as the fact that each step produces a square-free denominator and that the numbers can be summed in any order.

To prove that a prime-free span cannot occur in the pronic series, we would need to show that there is a mathematical principle or theorem that guarantees the presence of primes in every span. This could potentially be done through a proof by contradiction, assuming that a prime-free span exists and then showing that it leads to a contradiction.

In conclusion, while the evidence presented does suggest convergence of the series, further mathematical analysis and proof would be needed to fully support this claim. Additionally, proving that a prime-free span cannot occur would require a deeper understanding of the mathematical principles at play.