Hello,

When g=2762, g^2=7628644,

My distribution function,

pi(3*g/2)-pi(g/2) ~ pi(g^2)-pi((g-1)^2)

predicts 350 primes vs. actual 390 primes,

error= -40 or -10.2564%

I conjecture that g=2762 is the highest g for which the

deviation error is greater than 10%.

And I would that someone more skilled than I on this

list can search for a counterexample.

So far I've tested all g up to g=5250, and I had previously

tested all prime g up to ~17,000.

As I continue to suspect about this function that

the percentage of the error deviation grows progressively

smaller in amplitude, I began testing a range starting

g=25000 and I would further conjecture that the highest

g with percentage error > 8% will have occurred

prior to g=25000.

It was a theoretical scenario that brought me to test this

function in this neighborhood. I believe my theoretical

argument will make it clear why this phenomenon must

exist within the distribution of prime numbers.

> where the constant c = 3/2*ln(3/2)-1/2*ln(1/2) = 0.95477...

Be aware that my first formulation of

pi(3*g/2)-pi(g/2) ~ pi(g^2)-pi((g-1)^2)

may not be the most exact center for this

"order 1 order 2 codependancy"

within the distribution of primes, but it is close enough

that the percentage error goes to zero with increasing g.

I conjecture that one could consider

pi(3*g/2)-pi(g/2) ~ pi((g-1)^2)-pi((g-2)^2) or

pi(3*g/2)-pi(g/2) ~ pi((g+1)^2)-pi(g^2), for example

and these functions will also yield a percentage error that

goes to zero, maybe slower, maybe faster, somewhere there may

be an exact center (error drops fastest).

As I continue to suspect about this function that

the percentage of the error deviation grows progressively

smaller with increasing g.

I began testing a range starting

g=25000 and now I further conjecture that the highest g

for which the percentage error exceeds 8% will have occurred

prior to g=25000.

Here's as far as I got from g=25,000. The

highest percentage error found is < 4% in the tests below.

The sign of the error continues to change frequently

and the percentage of error continues to average

lower & lower.

Can someone please verify some of these numbers for me?

Thanks,

-Dick Boland

Data for g>25000

g g^2 PRED. ACT. ERROR count and %deviation

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25000 625000000 2476 2431 45 1.8510900863842040312

25001 625050001 2477 2475 2 0.080808080808080808

25002 625100004 2477 2421 56 2.3130937629078893018

25003 625150009 2477 2472 5 0.2022653721682847896

25004 625200016 2477 2465 12 0.4868154158215010141

25005 625250025 2478 2465 13 0.527383367139959432

25006 625300036 2478 2439 39 1.5990159901599015989

25007 625350049 2478 2470 8 0.3238866396761133602

25008 625400064 2478 2390 88 3.68200836820083682

25009 625450081 2478 2503 -25 -0.9988014382740711146

25010 625500100 2478 2489 -11 -0.4419445560466050622

25011 625550121 2478 2480 -2 -0.0806451612903225806

25012 625600144 2479 2466 13 0.5271695052716950526

25013 625650169 2479 2497 -18 -0.7208650380456547856

25014 625700196 2479 2483 -4 -0.1610954490535642368

25015 625750225 2479 2473 6 0.2426202992317023857

25016 625800256 2479 2468 11 0.4457050243111831442

25017 625850289 2479 2428 51 2.1004942339373970345

25018 625900324 2479 2428 51 2.1004942339373970345

25019 625950361 2479 2467 12 0.4864207539521686258

25020 626000400 2480 2466 14 0.5677210056772100567

25021 626050441 2480 2470 10 0.4048582995951417003

25022 626100484 2480 2453 27 1.1006930289441500203

25023 626150529 2480 2487 -7 -0.2814636107760353839

25024 626200576 2479 2493 -14 -0.5615724027276373846

25025 626250625 2480 2429 51 2.0996294771510909839

25026 626300676 2480 2465 15 0.6085192697768762677

25027 626350729 2480 2492 -12 -0.4815409309791332263

25028 626400784 2480 2400 80 3.3333333333333333333

25029 626450841 2480 2516 -36 -1.4308426073131955484

25030 626500900 2480 2512 -32 -1.2738853503184713375

25031 626550961 2480 2520 -40 -1.5873015873015873015

25032 626601024 2481 2490 -9 -0.3614457831325301204

25033 626651089 2482 2471 11 0.4451639012545528126

25034 626701156 2482 2486 -4 -0.1609010458567980691

25035 626751225 2482 2489 -7 -0.2812374447569304941

25036 626801296 2481 2426 55 2.2671063478977741137

25037 626851369 2481 2510 -29 -1.1553784860557768923

25038 626901444 2481 2448 33 1.3480392156862745097

25039 626951521 2481 2456 25 1.0179153094462540716

25040 627001600 2481 2469 12 0.4860267314702308626

25041 627051681 2482 2486 -4 -0.1609010458567980691

25042 627101764 2482 2472 10 0.4045307443365695792

25043 627151849 2482 2477 5 0.2018570851836899474

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