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So, no, 31 has not been overlooked.

Many years ago I had asked "Has 31 been overlooked?". On March 6, 2009 Steve Homewood replied with his newly discovered formula "So, no, 31 has not been overlooked." It was a real surprise. I had expected the solution to the Alhpa31 problem to take longer to be solved. My 2004 prediction (guess?) stated that the discovery would happen by 2014 so my expectation was that the discovery would happen in late 2013. What a pleasant surprise to all that follow the progress of my proof of the existence of God.

Steve Homewood of Harlow in the United Kingdom made this ingenious discovery. His success is awe inspiring. I do not know how many more mathematicians attempted to solve the A31 problem but I suspect a few including a feeble and failed attempt on my part. It is an accomplishment that Leonhard Euler and Stanislaw Ulam would have been proud of.

Homewood not only discovered that 31 in a simple polynomial can be used to generate prime numbers at a high success rate but found the missing link in the proof of the existence of God. From the early days of drafting Document A there was the 41 and 59 that fit but the 31 that should fit but had no formula that worked. I made a bold prediction and hoped some one would take up the challenge and find the productive prime number generator formula using the number 31.

Euler's formula 41 + n^2 + n generates primes 47.5 percent of the time. Ulam's formula 59 + 4 n^2 + 4n is productive at the rate of 43.7 percent. Homewood's formula 31 + 2n^2 + 4n is 43.5% correct for answers up to ten million. The formulae are added to the oddities list because they are simple, elegant and highly productive.

How important is solving the A31 problem? Well, I would say it is extremely important. All theories must have predictions. When the prediction comes true it helps confirm the thesis or theory. In this case 31 accomplished that. Now right at the start of pi (3.14159...) there are three numbers that can be used to generate prime numbers: 31, 41 and 59. These three numbers when combined 314159 become a prime number and even more odd the reverse 951413 is also a prime. There are more oddities in detailed here.

Even without Document A and the Proof of the Existence of God a simple polynomial prime number generator formula that is successful 43.5% is in itself important to the branch of mathematics called Number Theory and in it a sub-branch prime number "madness". I hope that his discovery will be met with as great enthusiasm he deserves and has received from me.

Vas Gardiakos

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From: Steve Homewood [mailto:stevehomewood@yahoo.co.uk]
Sent: Friday, March 06, 2009 2:06 PM
To: info@artmusicdance.com
Subject: 314

I thought you and your readers would be interested to know that there is a formula for 31, just like 41 and 59. It is 31 + 2n squared + 4n. It gives a result of primes 43.5% of the time for all answers up to 10,000,000. So, no, 31 has not been overlooked.

Steve Homewood

Hi Steve,

I really appreciate you taking the time to write to me. Did you come up with this formula or is it a published one? Are there other numbers besides 31, 41 and 59 using a simple or similar method that generate prime numbers?

On page http://www.artmusicdance.com/vaspi/questions.htm

2. When or if Alpha31 is confirmed it will strengthen the proposition. 31 must yield 39.9% or better to be considered as a valid candidate which can generate primes.

and K. A31 at

http://www.artmusicdance.com/vaspi/highlights.htm#A31

So 43.5% qualities.

On http://www.artmusicdance.com/vas/predictions.htm

I had predicted in 2004 that Alpha31 will be discovered by 2014. So thanks to you my prediction came in early.

I would like to hear more from you if you have the time. I will publish out dialog on my site. If you like I can add your website or blog or other link directed to you.

Thanks again.

Vas Gardiakos

Hi Vas,

After reading your website I put all my effort into finding the missing formula for 31. After finding it I checked the Internet to see if anyone else had already discovered it but no one has as far as I can tell. I will try and get back to you when I can and thank you for your website which sparked off my own interest,

Steve Homewood

Hello Steve,

I had scoured the web for 31 but also had no luck finding it used to generate primes.

I am very grateful for your time and effort to finding the missing formula for 31.

So I am pleased that 31 + 2n squared + 4n generates primes 43.5% of the time above the

39.9% minimum threshold I had specified.

What do you make that the formulas look similar especially 31 and 59?

31 + 2n squared + 4n

41 + n squared + n

59 + 4n squared + 4n

Shortly I will be revising the relevant pages to reflect your discovery.

Please tell me about what is your own interest that was sparked. You have

aroused my curiosity.

Vas Gardiakos

Hello Vas,

I had noticed that 31 and 59 both contained the '+ 4n' part of the equation and maybe this is what is needed to generate 40% primes in other formulas. More research needs to be done to confirm this of course. I have always had an interest in prime numbers but finding your site and linking it to pi was what sparked my interest for the search of the missing formula for 31, which proved to exist.

Steve Homewood

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Text related to A31 as it appeared just prior to March 6, 2009

I edited the text on several pages on this site to reflect Steve Homewood's discovery of the 31 prime number generator for easier reading. The five text boxes below are copied here to document how it appeared prior to the discovery and the editing.

http://www.artmusicdance.com/vaspi/highlights.htm

Proof of the Existence of Gods

2) In pi, 31, 41 and 59 are primes (3.1 41 59...). No formula has been discovered that will produce all the prime numbers however 41 + n squared + n and 59 + 4n squared + 4n are two of the most productive formulas known to yield prime numbers. Both 41 and 59 are used. Has 31 been overlooked? After all 314159 is a prime. See 5) bellow.

The great mathematician's Leonhard Euler's formula 41 + n squared + n turns out to be surprisingly good for large values of n. A computer showed that for the primes under ten million, the formula generated primes 47.5 percent of the time. The formula works even better for lower values of n. For values of n under 2,398, there's' an even chance of getting a prime. And for values under 100 the formula yields 86 primes and only 14 composite or non-prime numbers.

For n=0, the formula yields the prime number 41. For n=2, the prime number 47. Indeed, as n takes on successive values from 0 to 39, Euler's formula yields nothing but primes.

Stanislaw Ulam, a well known and respected mathematician and his coworkers discovered another prime generating formula that is almost as good as Euler's. The formula 59 + 4 n squared + 4n has a success rate of 43.7 percent.

http://www.artmusicdance.com/vaspi/highlights.htm

K. A31

Perhaps some time in the future an energetic mathematician will find that 31 can be used in a formula to generate prime numbers in similar fashion and as productively as Euler's and Ulam's formulas. Euler's formula using 41 and Ulam's using 59 yield prime numbers 47.5% and 43.7% respectively for primes up to ten million. For 31 to qualify as an equal to 41 and 59 it must generate at least 39.9% prime numbers. This 39.9% is calculated by 47.5 - 43.7 = 3.8. To establish the lowest acceptable level, 3.80 is subtracted from 43.7 to get the 39.9% figure.

http://www.artmusicdance.com/vaspi/questions.htm

Document A - Your Questions Answered by Vas

Question: What are the ways to strengthen or weaken the proposition that God numerated pi, the square root of pi and S?

Vas: Here are a few ways to help determine the validity that God numerated the three constants as illustrated in Document A.

2. When or if Alpha31 is confirmed it will strengthen the proposition. 31 must yield 39.9% or better to be considered as a valid candidate which can generate primes.

3. If some human can numerate pi more artfully than God this will weaken the proposition.

http://www.artmusicdance.com/vas/predictions.htm

The Gardiakos Oracle

2004: Alpha31 will be discovered by 2014. Explanation: In pi, 3.14159 (31 41 59) 41 and 59 can be used to generate prime numbers. No formula has been discovered that will produce all the prime numbers however two formulas that include 41 and 59 are very productive. Has 31 been overlooked in the search for other productive numbers?

http://www.artmusicdance.com/vaspi/questions.htm

Document A
Your Questions Answered by Vas

Vas: Here are a few ways to help determine the validity that God numerated the three constants as illustrated in Document A.

1. It will hurt the proposition if it was shown that most or even many random numbers have as many simple oddities as pi, the square root of two and S. It will help the proposition though more difficult to prove that random numbers generally do not have oddities or are not as interesting as the one illustrated in Document A. It will help the proposition if the three randomly chosen numbers do not have interweaving oddities.

2. When or if Alpha31 is confirmed it will strengthen the proposition. 31 must yield 39.9% or better to be considered as a valid candidate which can generate primes.

3. If some human can numerate pi more artfully than God this will weaken the proposition.

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