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General discussion :
Is there a 'probability of finding a prime' listed anywhere?
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Dad Send message
Joined: 28 Feb 18 Posts: 284 ID: 984171 Credit: 182,080,291 RAC: 0

Just wondering?
I'm fairly new to PrimeGrid and have approx 5 bronze and 1 Turquoise badge. I was just wondering if there is an 'overall' probability of finding a prime listed for each subproject.
EG: 321 Prime search AVG 1 prime per 52,345 credit, Proth Prime search AVG 1 prime per 1,234 credit
It would give me some idea of how I'm tracking
(I also realise that a prime could be found with 1 credit or with 1 billion credits, but the average would be nice to know)
Thanx  

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 13650 ID: 53948 Credit: 285,549,128 RAC: 40,408

The larger a number is, the harder it is to find a prime. The difficulty of finding a prime is roughly proportional to X^3 * ln(X), where X is the number of digits in the prime. So it's about 3000 times harder to find a million digit prime than it is to find a prime with one hundred thousand digits. This takes into account both the fact that larger primes are much rarer as well as taking much longer to run each individual test.
While I've never expressed it in terms of credit, on the smaller primes we do sometimes check the odds of a single test being prime.
The four smallest of our primality testing projects are, in order, GFN15, SGS, PPSE, and GFN16. As of last month, the odds of finding a prime in PPSE was about 1 in 14 thousand tests, and the odds of finding a GFN16 were about 1 in 16 thousand tests. Those numbers are approximately 450 thousand and 500 thousand digits long, respectively.
Let's say your computer can do a PPSE test in 5 minutes. On average, then, you could expect to find one PPSE prime after about 48 days of testing.
We can extrapolate those odds for, say the ESP project. ESP numbers are about 7 times larger than PPSE, so it should take 7^3 * ln(7) times as long to find an ESP prime. That's 667 times longer, or 32037 days or about 88 years.
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Dad Send message
Joined: 28 Feb 18 Posts: 284 ID: 984171 Credit: 182,080,291 RAC: 0

Thanx for the explanation Michael.
You say you don't express in 'credits', you use 'tests' instead, are 'tests' equivalent to WU's?
Thanx again  

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 13650 ID: 53948 Credit: 285,549,128 RAC: 40,408

Thanx for the explanation Michael.
You say you don't express in 'credits', you use 'tests' instead, are 'tests' equivalent to WU's?
Thanx again
Yes.
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My lucky number is 75898^{524288}+1  

axnVolunteer developer Send message
Joined: 29 Dec 07 Posts: 285 ID: 16874 Credit: 28,027,106 RAC: 0

So it's about 3000 times harder to find a million digit prime than it is to find a prime with one hundred thousand digits.
<snip>
ESP numbers are about 7 times larger than PPSE, so it should take 7^3 * ln(7) times as long to find an ESP prime. That's 667 times longer, or 32037 days or about 88 years.
FYI, this is not how logs work. ln(x)/ln(y) <> ln(x/y). Of course, the answer is still in the right ball park (since the cube term is the dominant one).
[There is a bigger issue, in that, the size of the numbers are not fixed  they grow. Hence the odds keep getting worse. Particularly severe for "sparse" projects like the conjecture ones]  

mackerelVolunteer tester
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Joined: 2 Oct 08 Posts: 2536 ID: 29980 Credit: 494,291,084 RAC: 3,328

The four smallest of our primality testing projects are, in order, GFN15, SGS, PPSE, and GFN16. As of last month, the odds of finding a prime in PPSE was about 1 in 14 thousand tests, and the odds of finding a GFN16 were about 1 in 16 thousand tests. Those numbers are approximately 450 thousand and 500 thousand digits long, respectively.
Time for some statistical fail on my part. Are the odds above calculated from the earlier formula, or worked out from actual testing?
If we use the general formula, would sieving affect the practical outcome? Say you have a test range of candidates (presieve), you could work out how many primes to expect in that range. After sieving, you would have the same number of expected primes, but a much shorter candidate list.
I didn't take statistics courses when offered as I thought I'd suck at it. Or do I suck at it because I didn't take those courses? Self fulfilling prophecy.  


ESP numbers are about 7 times larger than PPSE, so it should take 7^3 * ln(7) times as long to find an ESP prime. That's 667 times longer, or 32037 days or about 88 years.
FYI, this is not how logs work. ln(x)/ln(y) <> ln(x/y). Of course, the answer is still in the right ball park (since the cube term is the dominant one).
Right! If it goes as X^3 * ln X, it would be
(7X)^3 * ln (7X) = 7^3 * X^3 * (ln X + ln 7)
Since the ln 7 term will be negligible, it is more like 7^3 = 343 times slower.
/JeppeSN  

Scott BrownVolunteer moderator Project administrator Volunteer tester Project scientist
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Joined: 17 Oct 05 Posts: 2270 ID: 1178 Credit: 11,772,852,856 RAC: 13,185,936

[There is a bigger issue, in that, the size of the numbers are not fixed  they grow. Hence the odds keep getting worse. Particularly severe for "sparse" projects like the conjecture ones]
I have often wondered about this in the sense that conjecture projects probably don't follow the same pattern as others. More specifically, assuming that the base of users doesn't change drastically (a safer assumption on conjectures where many users have particularly dedicated interests in such projects), as primes are found within a conjecture (thereby often reducing the number of candidates to be tested substantially), I suspect that the odds getting worse as primes grow is actually less severe than in nonconjecture projects (or at least follows a far less smoothed growth and potentially could even experience very slight, brief declines...at least in application if not in theory).  

CrunchiVolunteer tester
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Joined: 25 Nov 09 Posts: 3070 ID: 50683 Credit: 63,378,402 RAC: 148

Scott, I think that primes in CRUS bases also follow this rule, but since all prime searching stopped in then time when prime is found that is difficulty to prove.
On the other hand: I found primes on CRUS bases when it wasn't expected at all, and also cannot find prime in other bases where I should ( by prediction) find prime long time ago. So primes follows first rule: nobody doesn't know position ( where prime is located)
Also I found big difference in predicted number of primes, and real number in nearrepdigit primes. But once again: it follow first rule :)
So CRUS maybe look "different" then other bases, but in real life it is just little more different then any other sequence we all search.
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General discussion :
Is there a 'probability of finding a prime' listed anywhere? 