User talk:Michael G. Kaarhus

Comments on PDF

These are my thoughts on your PDF "Twin Prime Conjectures I and II". I thought it would be good to move the discussion from [1] where it is not entirely relevant.

Your Lemmata 1-4 are really conjectures which together make up Conjecture II. Lemma 4 is trivially true (by the definition of g and q). Lemma 3 is true, as you claim; the finitude is implicit in the script I wrote to find terms of this sequence. But the proof is wrong: r need not be a factor of b#/2, as the example

$4\cdot 83=11\#/2-823$

demonstrates. Lemma 1 alone suffices to prove the Twin Prime Conjecture.

Lemma 2 has two parts: each r > 5 has a unique g and each r > 5 has a unique b. I haven't tested either and I'm not sure how likely they are.

Charles R Greathouse IV 23:48, 23 October 2012 (UTC)

Re: Comments on PDF

I am glad you mentioned lemma 3, because I explained it rather poorly and confusingly. Taking your example,

$4\cdot 83=11\#/2-823$

$4\cdot y=b\#/2-r$

$y=b\#/8-r/4$

I was trying to show that, if r is a factor of b#/2, and if y is integral, y will have a factor of r. That is, y will not be a g. The top equation is nowhere near approaching that point. However, 83 is not the only g linked to the pair (821, 823). There is at least one more linked g (the one listed in my PDF), and most likely others I didn't bother to calculate. But there cannot be infinitely more. We might find more g linked to 823 by incrementing the primorial, and recalculating. But the last possible linked g is this one:

$4\cdot y=821\#/2-823$

If that y is not integral (it isn't), or not prime, then it will do no good to increment the primorial again and recalculate. If you increment the primorial to 823#, then (if y is integral), y will have a factor of 823 (which is r). Testing my prediction:

(pfact(823)/8) - (823/4)

259564192498446760735885335934272453568013110814328101747548055321 821669958995637304035693589931387023234066532008293790316510069221 809264423082811821307582136523992998691260242224616723884631524687 285421815196211150982185776109507210893904337289129652647591392756 282463561376123757085665516732407762306891370512351980006762437920 78084680578

./823

315387840216824739654781696153429469705969757976097328976364587268 313086219921795023129639841957942920089995786158315662596002514242 781609262555056891017718270381522477146124231135621778717656773617 600755547018482564984429861615440110442168089051190343435712506386 734463622571231782607126994814590233665724630027159149461436741094 50892686

(The ./823 means "answer/823")

823#/8 - 823/4 is an even integer. But it also happens to have a factor of 823 (the second large integer). If you increment the primorial beyond 823#/2, the new primorial (even though divided by 2) will still have a factor of 823. And the new y (if integral) will also have a factor of 823. I know this is true, but I haven't yet figured out exactly how to prove it.

Because it contains so many errors, I have temporarily deleted my PDF (conj-ii.pdf) from my website. I am in the process of overhauling it as my schedule and my slowness permit (Math is not my day job). I will simplify it, and probably not use lemmata.

I have debugged my perl script so that it returns only prime g, and leaves me a really long list of potential r that are too large for it to test for primality. It's a long, byzantine and complex script. Unfortunately, it calculates up to only 47#, so it will miss any small 8g that sum with some large 2r to equal some primorial larger than 47#.

Thank you for reviewing my PDF, making suggestions and correcting my sequence.

Michael G. Kaarhus 03:23, 27 October 2012 (UTC)

OK. Now I can prove Lemma 3

Michael G. Kaarhus 11:48, 28 October 2012 (UTC)

My latest version of Twin Prime Conjectures I and II is now online at http://www.christaboveme.com/pri/conj-ii.pdf

Michael G. Kaarhus 02:20, 30 October 2012 (UTC)

Well it looks like, in light of the revision, my comments about the lemmas are now moot. I still feel that the claims of Conjecture II should be separated. On one hand you have the conjecture that A218046 is infinite, which (although implying the Twin Prime Conjecture) is not too strong. On the other hand you have the claim that for each twin prime there is a unique associated g, which seems like a quite strong claim to me. (If I had more time I would search for a counterexample.)

I should mention that the "Chen-like" result discussed in Appendix A is most likely an artifact of the law of small numbers. You're looking at numbers which are relatively prime to 6, and so there aren't enough small prime factors left for the numbers to other than prime or semiprime with much probability. To make that slightly less handwavy: the probability that a random integer near x and relatively prime to 6 is prime or semiprime is roughly (if I do my calculations correctly) (13/6 + log log x)/log x, which is large for the sizes of numbers considered in the appendix.

Similar remarks apply to Appendix B.

The (very) simple theorem in Appendix C is correct, though the last parenthetical should really be (b >= 5) for those inclined to use nonprime (or, indeed, noninteger) values of b.

You might consider for your Perl script implementing a simple probable-primality test like Fermat's. This way you can test large numbers more quickly, leaving the final proof of primality to some more specialized tool like Primo.

On an unrelated but relevant point: you did a very nice job TeXing the paper, it looks nice.

Charles R Greathouse IV 20:39, 30 October 2012 (UTC)

I split Conjecture II into two conjectures, changed to b >= 5 for the simple theorem, and specified which type of primorial I use throughout. I also made tables for lists of g that would fit into tables. I also let LaTex number and format my conjectures, theorems, and definition, instead of doing that myself. I am unable you use your excellent calculations from probability, because I don't understand them. I have not learnt probability or number theory, and my knowledge of calculus is both sketchy and rusty.

The new version has a different title, but is at the same URL:
http://www.christaboveme.com/pri/conj-ii.pdf
Thank you for the suggestions and comments.
Michael G. Kaarhus 11:34, 1 November 2012 (UTC)

your new TP paper

I am curious if you have/will submit as an OEIS sequence (either p or Y)? very interesting ... ref: A--Bill McEachen 13:39, 9 November 2013 (UTC)

well I wrote Pari code for your sequence. I can supply that as well as data but I believe your "1st 27" list is incorrect. I believe 13931 is actually the 60th entry. My list is 3,5,11,17,29,71,131,197,269,359,479,491,647,1277,1289,1367,1607...(I computed the 1st 288 terms). I like sequences with a high "multiplier" on generating primes, and yours does this.--Bill McEachen 00:33, 10 November 2013 (UTC)

I have not submitted my results as an OEIS sequence. After I published my article "Twin Prime Quads Family II", I began to wonder how noteworthy my equation is. Consider the lines $y=x$, $y=2x$, $y=3x$, and $y=6x$. Each contains all the averages of all the twin prime pairs that exist. So, how surprising is it that we are able to find quadratics that contain or intersect twin prime averages? Maybe it's not very remarkable.

The equation in my article is $(x-h)*(x+h+3)/2$. On checking my work, I find that $(x-h)*(x+h+3)$ also generates averages of twin prime pairs. More evidence that maybe equations that generate twin prime averages are rather common. Maybe no one except a few people have tried to find such equations, and so they are unusual, just because no one does this. Do you think it is worth submitting?

I found an error in the file, fam2-h1.txt. It gives solutions for $(x^{2}+3x-4)/2$,

but it says they are solutions for $x^{2}+3x-4$. I neglected to type the /2. I will have to correct that. I did not find any errors in my article. Maybe we are using slightly different equations? The script I used was

/* wxMaxima */ x:0$ y:0$ e:0$ f(x):= ((x-1)*(x+4)/2)$

for x:1 thru 6000 do(y:f(x), if (primep(y-1) and primep(y+1)) then

(e:e+1, print(e,",",x,",",y)))$

print("program_done") /* Michael Kaarhus, Nov 9, 2013 */$

It generates the TP averages

$(12,18,42,102,150,228,348,432,462,858...)$

When set to f(x):= ((x-1)*(x+4)), the same script generates TP averages

$(6,150,1050,3300,10500,11550,43050,71550,97650,100800...)$

which is a higher multiplier. And they all end in 0, except for the first term. Am I doing something wrong? I should be able to get

$(4,6,12,18,30,72,132,198,270,360,480,492,648,1278,1290,1368,1608...)$

which is your sequence above with 1 added to each term. But that's not the output I get.

Thank you for reading my article and sharing your findings. Michael G. Kaarhus 06:07, 10 November 2013 (UTC)

I was speaking to the eq'n p^2 -(p+2) only in http://www.christaboveme.com/pri/newprime.pdf. I only now see your later May paper.

well, certainly p=3 should generate a hit, as it produces (3,5). Also, 131 gives (17027,17029).etc your code above is for a different eq'n.

also, I must ensure you saw Rivera's puzzle post: Puzzle44

Ah. I probably used a buggy perl script to find solutions for $p^{2}-(p+2)$ from a list of twin primes. I will write a maxima script for it, and correct the article. I do not have anything that will run a PARI or a mathematica script. In the meantime, I have temporarily pulled that article from my site.

Regarding Rivera's puzzle, I see I am not the only one who does this kind of thing. My reason for writing newprime.pdf was to work toward demonstrating that, for some twin primes p, you can always find larger twin primes, which suggests that the twin primes are infinitely many.

Thanks again for checking my work. Michael G. Kaarhus 13:23, 10 November 2013 (UTC)

---

I checked, and found that my "1st 27" list is correct, except that I omitted 3. Checking your list, I find that (59, 479, 491, 647) are primes, but are not twin primes. My sequence is twin primes p, such that $p^{2}-p-2$ is the average of a larger twin prime pair. Michael G. Kaarhus 17:59, 11 November 2013 (UTC)