Ratio between near-WW primes

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I was wondering if the probability for near-WW primes is linear with A. It feels like it should be (which doesn't say much) and from the findings so far it could be true: 8 for |A|<=100 and 98 for |A|<=1000.

If it is, we could very well find a |A|=1 since we're at 10 % done.
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For a prime, say p = 1212976000000000003 (taken as any prime near the current leading edge), there are 1212976000000000003 different values of A which we would expect (if we do not know anything else) to be equally probable. These are:

-606488000000000001 -606488000000000000 -606487999999999999 -606487999999999998 ... -3 -2 -1 0 +1 +2 +3 ... +606487999999999998 +606487999999999999 +606488000000000000 +606488000000000001

There are 1800 values with 100 < |A| <= 1000.
There are 180 values with 10 < |A| <= 100.
There are 20 values with 0 < |A| <= 10.
There is 1 value with A = 0.

So in the long run, we expect the counts in the "categories" to be in the ratio 1800 : 180 : 20 : 1.

Just one single prime in the A = 0 category will earn eternal fame to PrimeGrid. Statistically, expect about 1 such prime when we have 1800 primes in 100 < |A| <= 1000.

/JeppeSN

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Yves Gallot
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The probability for near-primes is linear with A.

The ending point for PrimeGrid's prior Wieferich search on PRPNet is 6.07132e17 and for Wall-Sun-Sun search is 2.6438336e17.

If p = 12e17 (the current leading edge),
the expected number of Wieferich near-primes with |A| <= 1000 is (log(log(12e17)) - log(log(6.07132e17))) * 2001 ~ 33
the expected number of Wall-Sun-Sun near-primes with |A| <= 1000 is (log(log(12e17)) - log(log(2.6438336e17))) * 2001 ~ 74

28 Wieferich and 70 Wall-Sun-Sun near-primes were found with |A| <= 1000.

For |A| <= 10, the expected number of Wieferich and Wall-Sun-Sun near-primes is (33 + 74) / 100 ~ 1.

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So far the results match nicely with 13 and 130. I'm not so sure it holds when it comes to A = 0.

We don't look at the actual remainder, but at A = remainder/p, right? So it accomodates for the decreasing prime density at increasing size of the numbers. Thus my feeling is, A = 0 = remainder, will be much less probable and it's probability decreases with increasing size of the number - same as the prime density decreases.

Even more so, if 1800 for |A|<=1000 is really sufficient to produce a Wieferich or WSS prime, then we are 8% there? That seems way too easy to me. If 1.5E18 is 8%, we'd find one within 2E19.

Also I read the density of Wieferich primes is log(log(n)) for 1 to n. Given that 2 are already known, we'd need to go up to 10^1000 to have a good chance to find another one. Even if we chalk one of them up to pure chance, another would be due only within 1...10^100. Far beyond our reach.

Or did I get you wrong?
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When p (the primes we search) is doubled, the probability in each category (100 < |A| <= 1000, etc.) is halved. So yes, it becomes harder when p grows.

But we think the proportion 1800 : 180 : 20 : 1 should stay the same, as p grows.

The log(log(n)) heuristics is derived from thinking exactly in this way, I would say. Note the following (natural logarithms):
log(log(1.5e+1)) = 1
log(log(1.6e+3)) = 2
log(log(5.3e+9)) = 3
log(log(5.1e+23)) = 4
log(log(2.9e+64)) = 5

So we should find about 1 Wieferich (plus 1 Wall-Sun-Sun) between 5.3e+9 and 5.1e+23. And find the same between 5.1e+23 and 2.9e+64. So this shows how they are supposed to get rarer.

But it should not be "memorizing", so given that the leading edge is near exp(exp(3.7)) right now, no matter how many have been found up to now, we should expect to find about 1 Wieferich (and 1 Wall-Sun-Sun) between now and exp(exp(1 + 3.7)) = 5.6e+47. But you see, we are not able to search that far. So from today and on, we have to be extremely lucky to find anything.

In an old post, I made a graph, I think I can find it.

(Of course, we have no proof these heuristics are correct. In theory, Wieferich primes could be frequent or more rare.)

/JeppeSN

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(Of course, we have no proof these heuristics are correct. In theory, Wieferich primes could be frequent or more rare.)

And WSS primes may not exist if A = 0 is impossible for a currently unknown reason.

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Me:

There are 1800 values with 100 < |A| <= 1000.
There are 180 values with 10 < |A| <= 100.
There are 20 values with 0 < |A| <= 10.
There is 1 value with A = 0.

So in the long run, we expect the counts in the "categories" to be in the ratio 1800 : 180 : 20 : 1.

Just one single prime in the A = 0 category will earn eternal fame to PrimeGrid. Statistically, expect about 1 such prime when we have 1800 primes in 100 < |A| <= 1000.

I just realied the stats page does not use:
100 < |A| <= 1000
10 < |A| <= 100
0 < |A| <= 10
A = 0

It uses:
0 <= |A| <= 1000
0 <= |A| <= 100
0 <= |A| <= 10
A = 0

So the ratio with that is simply 2001 : 201 : 21 : 1.

/JeppeSN

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edit: I only saw Jeppe's post about the stats page now, so I adjusted the number accordingly.

I made a table comparing the actual number of |A| in each category with the expected number, iff (oh yeah) 2001:201:21:1 is correct. I only re-distributed the number of near-WW to follow that ratio, this is by no means a calculation of the expected total number of near-WW.

Number of near-WW found: 201 Distribution: |A|<= | Actual | Expected -------|--------|---------- 1000 | 201 | - 100 | 24 | 20 10 | 1 | 2 0 | 0 | 0 Based on the current number of near-WW found, the probability for a WW or Wieferich prime is: 10 %. At 1.8e19 we would reach: 56 %. 100 % would be reached at: 3.2e19.

The "forecast" was made by taking the current upper bound of tested numbers.
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Update

Tests upper bound: 3.3e18 / 18e18 (18 %) Number of near-WW found: 205 Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 205 | - | - 100 | 24 | 21 | +14 % 10 | 1 | 2 | -50 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 10 %. At 18e18 we would reach: 53 %. 100 % would be reached at: 34e18.

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Update

Tests upper bound: 3.7e18 / 18e18 (21 %) Number of near-WW found: 214 Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 214 | - | - 100 | 25 | 21 | +19 % 10 | 1 | 2 | -50 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 11 %. At 18e18 we would reach: 54 %. 100 % would be reached at: 34e18.

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Update

Tests upper bound: 4.3e18 / 18e18 (24 %) Number of near-WW found: 224 Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 224 | - | - 100 | 26 | 23 | +13 % 10 | 1 | 2 | -50 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 11 %. At 18e18 we would reach: 46 %. 100 % would be reached at: 39e18.

What the prediction doesn't take into account is that the near-WWs are apparently getting more rare with increasing p.
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Update 26-04-2021

Tests upper bound: 4.6e18 / 18e18 (26 %, +2 %) Number of near-WW found: 234 (+10) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 234 | - | - 100 | 27 | 24 | +13 % 10 | 1 | 2 | -50 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 12 % (+ 1%). At 18e18 we would reach: 47 % (+1 %). 100 % would be reached at: 38e18 (-1e18).

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Update 03-05-2021

Tests upper bound: 4.8e18 / 18e18 (27 %, +1 %) Number of near-WW found: 236 (+2) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 236 | - | - 100 | 28 | 24 | +17 % 10 | 1 | 2 | -50 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 12 % (+ 0%). At 18e18 we would reach: 44 % (-3 %) 100 % would be reached at: 41e18 (+3e18)

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Update 10-05-2021

Tests upper bound: 5.1e18 / 18e18 (28 %, +1 %) Number of near-WW found: 240 (+4) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 240 | - | - 100 | 29 | 24 | +21 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 12 % (+ 0%). At 18e18 we would reach: 42 % (-2 %) 100 % would be reached at: 43e18 (+2e18)

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What the prediction doesn't take into account is that the near-WWs are apparently getting more rare with increasing p.

They are--so much so that the expected number of WW's in a range [a, b] is best modeled not as const * (b-a) but as log log b - log log a. For example, the previously unsearched range of WW is [2.63691e17, 5.14e18] for WSS and [5.97077e17, 5.14e18] for Wieferich, so one would expect about 0.0714 new WSS primes and 0.0513 new Wieferich primes, along with 143 and 103 near-finds (i.e. 2001x more) respectively. We've actually found 141 and 99 respectively, with two more Wieferich near-finds currently waiting for double-checkers. Pretty accurate!

Unfortunately, this means that we've already missed our best shot. The same formula predicts about 58 more near-finds of each of the two types below 2^64, with about 0.0292 actual finds of each type. So overall about a 6% chance of success.

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Thanks Ravi, is it still correct to estimate the expected number of near-WW and WW from the actual number of near-WW found? I.e. what I do with that table.

Update 17-05-2021

Tests upper bound: 5.4e18 / 18e18 (30 %, +3 %) Number of near-WW found: 247 (+7) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 247 | - | - 100 | 30 | 25 | +20 % 10 | 1 | 2 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 12 % (+ 0%). At 18e18 we would reach: 42 % (-0 %) 100 % would be reached at: 43e18 (+0e18)

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Yes, the table looks good to me.

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Update 03-25-2021

Tests upper bound: 5.6e18 / 18e18 (31 %, +1 %) Number of near-WW found: 250 (+3) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 250 | - | - 100 | 30 | 25 | +20 % 10 | 1 | 2 | -50 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 13 % (+ 1%). At 18e18 we would reach: 40 % (-2 %) 100 % would be reached at: 45e18 (+2e18)

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Update 06-01-2021

Tests upper bound: 5.7e18 / 18e18 (32 %, +1 %) Number of near-WW found: 251 (+1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 251 | - | - 100 | 30 | 25 | +20 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 13 % (+ 0%). At 18e18 we would reach: 40 % (-0 %) 100 % would be reached at: 46e18 (+e18)

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Update 06-07-2021

Tests upper bound: 5.9e18 / 18e18 (33 %, +1 %) Number of near-WW found: 255 (+4) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 255 | - | - 100 | 30 | 26 | +15 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 13 % (+ 0%). At 18e18 we would reach: 39 % (-1 %) 100 % would be reached at: 46e18 (+0e18)

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Update 06-14-2021

Tests upper bound: 6.0e18 / 18e18 (33 %, +0 %) Number of near-WW found: 256 (+1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 256 | - | - 100 | 30 | 26 | +15 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 13 % (+ 0%). At 18e18 we would reach: 38 % (-1 %) 100 % would be reached at: 47e18 (+1e18)

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Update 06-21-2021

Tests upper bound: 6.1e18 / 18e18 (34 %, +1 %) Number of near-WW found: 259 (+3) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 259 | - | - 100 | 30 | 26 | +15 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 13 % (+ 0%). At 18e18 we would reach: 38 % (+0 %) 100 % would be reached at: 47e18 (+0e18)

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Update 06-28-2021

Tests upper bound: 6.3e18 / 18e18 (35 %, +1 %) Number of near-WW found: 262 (+3) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 262 | - | - 100 | 30 | 26 | +15 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 13 % (+ 0%). At 18e18 we would reach: 37 % (-1 %) 100 % would be reached at: 48e18 (+1e18)

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Update 07-05-2021

Tests upper bound: 6.4e18 / 18e18 (36 %, +1 %) Number of near-WW found: 265 (+3) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 265 | - | - 100 | 30 | 27 | +11 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 13 % (+ 0%). At 18e18 we would reach: 37 % (-0 %) 100 % would be reached at: 48e18 (+0e18)

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Update 07-12-2021

Tests upper bound: 6.6e18 / 18e18 (37 %, +1 %) Number of near-WW found: 267 (+2) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 267 | - | - 100 | 30 | 27 | +11 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 13 % (+ 0%). At 18e18 we would reach: 36 % (-0 %) 100 % would be reached at: 50e18 (+0e18)

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Update 19-07-2021

Tests upper bound: 6.7e18 / 18e18 (37 %, +0 %) Number of near-WW found: 267 (+-0) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 267 | - | - 100 | 30 | 27 | +11 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 13 % (+- 0%). At 18e18 we would reach: 36 % (-0 %) 100 % would be reached at: 50e18 (+0e18)

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Update 26-07-2021

Tests upper bound: 6.8e18 / 18e18 (38 %, +1 %) Number of near-WW found: 268 (+1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 268 | - | - 100 | 30 | 27 | +11 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 13 % (+ 0%). At 18e18 we would reach: 36 % (-0 %) 100 % would be reached at: 51e18 (+0e18)

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Update 03-08-2021

Tests upper bound: 7.0e18 / 18e18 (39 %, +1 %) Number of near-WW found: 269 (+1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 269 | - | - 100 | 30 | 27 | +11 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 13 % (+ 0%). At 18e18 we would reach: 35 % (-1 %) 100 % would be reached at: 52e18 (+1e18)

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Update 09-08-2021

Tests upper bound: 7.1e18 / 18e18 (39 %, +0 %) Number of near-WW found: 270 (+1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 270 | - | - 100 | 30 | 27 | +11 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 14 % (+ 1%). At 18e18 we would reach: 34 % (-1 %) 100 % would be reached at: 53e18 (+1e18)

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Update 17-08-2021

Tests upper bound: 7.2e18 / 18e18 (40 %, +1 %) Number of near-WW found: 272 (+2) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 272 | - | - 100 | 30 | 27 | +11 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 14 % (+ 0%). At 18e18 we would reach: 34 % (-0 %) 100 % would be reached at: 53e18 (+0e18)

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Update 24-08-2021

Tests upper bound: 7.4e18 / 18e18 (41 %, +1 %) Number of near-WW found: 273 (+1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 273 | - | - 100 | 30 | 27 | +11 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 14 % (+ 0%). At 18e18 we would reach: 33 % (-1 %) 100 % would be reached at: 54e18 (+1e18)

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Update 31-08-2021

Tests upper bound: 7.6e18 / 18e18 (42 %, +1 %) Number of near-WW found: 274 (+1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 274 | - | - 100 | 30 | 28 | +7 % 10 | 1 | 3 | -67 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 14 % (+ 0%). At 18e18 we would reach: 32 % (-1 %) 100 % would be reached at: 56e18 (+1e18)

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A lot of finds this week (ok, 1.5 weeks) and also a WSS with A=-1!

Update 09-09-2021

Tests upper bound: 7.7e18 / 18e18 (43 %, +1 %) Number of near-WW found: 279 (+5) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 279 | - | - 100 | 31 | 28 | +11 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 14 % (+ 0%). At 18e18 we would reach: 33 % (+1 %) 100 % would be reached at: 55e18 (-1e18)

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Update 13-09-2021

Tests upper bound: 7.8e18 / 18e18 (43 %, +0 %) Number of near-WW found: 279 (+0) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 279 | - | - 100 | 31 | 28 | +11 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 14 % (+ 0%). At 18e18 we would reach: 32 % (-1 %) 100 % would be reached at: 56e18 (+1e18)

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Update 20-09-2021

Tests upper bound: 8.0e18 / 18e18 (44 %, +1 %) Number of near-WW found: 281 (+2) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 281 | - | - 100 | 31 | 28 | +11 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 14 % (+ 0%). At 18e18 we would reach: 32 % (-0 %) 100 % would be reached at: 57e18 (+1e18)

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Update 27-09-2021

Tests upper bound: 8.1e18 / 18e18 (45 %, +1 %) Number of near-WW found: 283 (+2) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 283 | - | - 100 | 31 | 28 | +11 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 14 % (+ 0%). At 18e18 we would reach: 31 % (-1 %) 100 % would be reached at: 57e18 (+0e18)

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Update 04-10-2021

Tests upper bound: 8.3e18 / 18e18 (46 %, +1 %) Number of near-WW found: 283 (+-0) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 283 | - | - 100 | 31 | 28 | +11 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 14 % (+- 0%). At 18e18 we would reach: 31 % (-1 %) 100 % would be reached at: 59e18 (+2e18)

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I could understand this if you said - percentage wise - how much worse the chances are getting.
Compared to first estimate.
Compared to recent.

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You can see that from the ever decreasing "At 18e18 we would reach:" value. It decreases because we find fewer and fewer near hits. Or did you mean something else?

Honestly, I'm not really convinced it makes sense as a forecast (likely it doesn't) since it doesn't take into account that these misses already happened and probability doesn't care what did occur. Just because we found no WWS prime with that many nears doesn't mean we're bound to find one with a specific probability.

Anyway, I think it's a nice statistic that shows how the subproject progresses and it also shows that the "nearness" appears to be statistically distributed as JeppeSN suggested.

Update 11-10-2021

Tests upper bound: 8.5e18 / 18e18 (47 %, +1 %) Number of near-WW found: 285 (+2) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 285 | - | - 100 | 31 | 28 | +7 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 14 % (+ 0%). At 18e18 we would reach: 30 % (-1 %) 100 % would be reached at: 60e18 (+1e18)

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Update 18-10-2021

Tests upper bound: 8.7e18 / 18e18 (48 %, +1 %) Number of near-WW found: 288 (+3) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 288 | - | - 100 | 31 | 28 | +7 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 14 % (+ 0%). At 18e18 we would reach: 30 % (-0 %) 100 % would be reached at: 60e18 (+0e18)

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We reached 2^63 of 2^64 tests. At first glance it looks like we're nearly done, but exponentials being counter-intuitive to many people (like me) and it's just half way done... ;)

Update 05-11-2021

Tests upper bound: 9.2234e18 / 18e18 = 2^63 / 2^64 (50 %, +1 %) Number of near-WW found: 299 (+4) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 299 | - | - 100 | 31 | 30 | +3 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 15 % (+ 1%). At 18e18 we would reach: 29 % (-1 %) 100 % would be reached at: 62e18 (+2e18)

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Update 09-11-2021

Tests upper bound: 9.3e18 / 18e18 (51 %, +1 %) Number of near-WW found: 301 (+2) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 301 | - | - 100 | 33 | 30 | +10 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 15 % (+ 1%). At 18e18 we would reach: 29 % (-0 %) 100 % would be reached at: 62e18 (+0e18)

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Update 15-11-2021

Tests upper bound: 9.4e18 / 18e18 (51 %, +0 %) Number of near-WW found: 302 (+1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 302 | - | - 100 | 33 | 30 | +10 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 15 % (+ 1%). At 18e18 we would reach: 29 % (-0 %) 100 % would be reached at: 62e18 (+0e18)

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Update 22-11-2021

Tests upper bound: 9.5e18 / 18e18 (52 %, +1 %) Number of near-WW found: 302 (+-0) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 302 | - | - 100 | 33 | 30 | +10 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 15 % (- 0%). At 18e18 we would reach: 29 % (-0 %) 100 % would be reached at: 63e18 (+1e18)

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A bit late...

Update 29-11-2021

Tests upper bound: 9.6e18 / 18e18 (52 %, +0 %) Number of near-WW found: 305 (+3) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 305 | - | - 100 | 34 | 31 | +10 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 15 % (- 0%). At 18e18 we would reach: 29 % (-0 %) 100 % would be reached at: 63e18 (+0e18)

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Things were slow. The GFN-challenge seems to have drained GPU power from WWSS. My 1660 Super will be back soon though, so expect a big speed-up next week.

Update 13-12-2021

Tests upper bound: 9.8e18 / 18e18 (53 %, +0 %) Number of near-WW found: 308 (+1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 308 | - | - 100 | 34 | 31 | +10 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 15 % (+ 0%). At 18e18 we would reach: 29 % (-0 %) 100 % would be reached at: 64e18 (+1e18)

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No new near-WWSSprimes this week. Not unexpected, since during the challenge the completion rate had dropped from 14k/day to 5k/day. It's back up again though, so unless many computers take a break over the christmas holidays, hopefully this week will fare better again.

Update 20-12-2021

Tests upper bound: 9.9e18 / 18e18 (54 %, +1 %) Number of near-WW found: 308 (+-0) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 308 | - | - 100 | 34 | 31 | +10 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 15 % (+- 0%). At 18e18 we would reach: 29 % (-0 %) 100 % would be reached at: 64e18 (+0e18)

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Update 03-01-2022

Last update ist two weeks back, so mind that when looking at the differences. And we reached 1e19. Only 8 quintillion numbers left!

Tests upper bound: 10.1E18 / 18E18 (55 %, +1 %) Number of near-WW found: 310 (+-2) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 310 | - | - 100 | 34 | 31 | +10 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 16 % (+ 1%). At 18e18 we would reach: 28 % (-1 %) 100 % would be reached at: 65E18 (+1E18)

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Cool. The rest of the discoveries we do, will have twenty digits, like 10017779178182939981 by Grzegorz Roman Granowski. /JeppeSN

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Update 10-01-2022

Tests upper bound: 10.2e18 / 18e18 (55 %, +0 %) Number of near-WW found: 311 (+-1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 311 | - | - 100 | 34 | 31 | +10 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 16 % (+ 1%). At 18e18 we would reach: 28 % (-0 %) 100 % would be reached at: 66e18 (+1e18) Last week average numbers tested per day: 1.41E16 (+- N/A) ETA project completion: 582 days (+- N/A)

Things are still slow, last week's average is 9400 completed tasks/day. Due to DC that corresponds to 14E16 numbers checked / day. At that rate it'll be 582 days to complete the project. I added that to the stats.
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582 days could be a best case therefore but if loads of people jump on it how dramatically could that come down? I'm playing safe & only parking at 1.1B credit - 2B would take me nearly all year but only assuming enough work!

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582 days could be a best case therefore but if loads of people jump on it how dramatically could that come down? I'm playing safe & only parking at 1.1B credit - 2B would take me nearly all year but only assuming enough work!

WW has been running for 380 days and the leading edge is 10,200 P.
Then the remaining time is 380 * (2^64/10200e15) - 380 = 307 days.
The Riemann's Birthday Challenge (17-20 September) could be the end of the project.

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The Riemann's Birthday Challenge (17-20 September) could be the end of the project.

It would be an unusual challenge if we ran out of work before the end of the challenge. /JeppeSN

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Yes, it's only the ETA based on last week's numbers. I clarified that in the stats.

Update 17-01-2022

Tests upper bound: 10.3e18 / 18e18 (56 %, +1 %) Number of near-WW found: 312 (+1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 312 | - | - 100 | 35 | 31 | +13 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 16 % (+ 0%). At 18e18 we would reach: 28 % (-0 %) 100 % would be reached at: 66e18 (+0e18) Last week average numbers tested per day: 13.3E15 (-0.8E15) ETA project completion (based on last week only): 609 days (+ 27 days)

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Update 24-01-2022

Tests upper bound: 10.4e18 / 18e18 (57 %, +1 %) Number of near-WW found: 313 (+1) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 313 | - | - 100 | 35 | 31 | +13 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 16 % (+ 0%). At 18e18 we would reach: 28 % (-0 %) 100 % would be reached at: 67e18 (+1e18) Last week average numbers tested per day: 12.2E15 (-1.1E15) ETA project completion (based on last week only): 657 days (+ 46 days)

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No new near-hit. High GPU prices are hurting innocent DC-project! The humanity!

Update 01-01-2022

Tests upper bound: 10.5e18 / 18e18 (57 %, +0 %) Number of near-WW found: 313 (+-0) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 313 | - | - 100 | 35 | 31 | +13 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 16 % (+-0%). At 18e18 we would reach: 27 % (-1 %) 100 % would be reached at: 67e18 (+1e18) Last week average numbers tested per day: 16.0E15 (+3.8E15) ETA project completion (based on last week only): 494 days (- 163 days)

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Update 07-02-2022

Tests upper bound: 10.5e18 / 18e18 (57 %, +0 %) Number of near-WW found: 313 (+-0) Distribution: |A|<= | Actual | Expected | Deviation -------|--------|----------|----------- 1000 | 313 | - | - 100 | 35 | 31 | +13 % 10 | 2 | 3 | -33 % 0 | 0 | 0 | 0 % Based on the current number of near-WW found, the probability for a WW prime is: 16 % (+-0%). At 18e18 we would reach: 27 % (-0 %) 100 % would be reached at: 67e18 (+0e18) Last week average numbers tested per day: 8.6E15 (-7.4E15) ETA project completion (based on last week only): 922 days (+ 428 days)

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Just curious if Bur might of forgotten to update this sub project stats. Hopefully everything is OK for him.

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Greetings ... will there be an update ... regarding WW ?

the last was over 2 months ago !

regards, Grzegorz Roman Granowski
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I think he might just have gotten tired of this or forgot-he was last online 10 days ago on mersenneforum.
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The leading edge is 11,706.982 P (visible on http://www.primegrid.com/server_status_subprojects.php), which means that the project is just over 65% now.
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The leading edge is 11,706.982 P (visible on http://www.primegrid.com/server_status_subprojects.php), which means that the project is just over 65% now.

I am confused if this tells us how much the project is finished on an on-going basis.
How, please, can I work out how much is left to be done?
The rate this has picked up recently, WW may not be a viable challenge in September.
(Really I am worried about not getting the next badge)

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The leading edge is 11,706.982 P (visible on http://www.primegrid.com/server_status_subprojects.php), which means that the project is just over 65% now.

I am confused if this tells us how much the project is finished on an on-going basis.
How, please, can I work out how much is left to be done?
The rate this has picked up recently, WW may not be a viable challenge in September.
(Really I am worried about not getting the next badge)

The leading edge is now 12,612.271P. The limit of the software for project is 18,000P, so that is when the project will end. So we are at 70% now.

For reference, each work unit advances the project by .003P, so my 3070 is advancing the project by about .270P each day.
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The leading edge is 11,706.982 P (visible on http://www.primegrid.com/server_status_subprojects.php), which means that the project is just over 65% now.

I am confused if this tells us how much the project is finished on an on-going basis.
How, please, can I work out how much is left to be done?
The rate this has picked up recently, WW may not be a viable challenge in September.
(Really I am worried about not getting the next badge)

The leading edge is now 12,612.271P. The limit of the software for project is 18,000P, so that is when the project will end. So we are at 70% now.

For reference, each work unit advances the project by .003P, so my 3070 is advancing the project by about .270P each day.

I like to use the force in these calculations - oh crap - I won't get the next badge.
But I am going to go for it.
It is worthwhile - to try.

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The leading edge is 11,706.982 P (visible on http://www.primegrid.com/server_status_subprojects.php), which means that the project is just over 65% now.

I am confused if this tells us how much the project is finished on an on-going basis.
How, please, can I work out how much is left to be done?
The rate this has picked up recently, WW may not be a viable challenge in September.
(Really I am worried about not getting the next badge)

The leading edge is now 12,612.271P. The limit of the software for project is 18,000P, so that is when the project will end. So we are at 70% now.

For reference, each work unit advances the project by .003P, so my 3070 is advancing the project by about .270P each day.

I like to use the force in these calculations - oh crap - I won't get the next badge.
But I am going to go for it.
It is worthwhile - to try.

Ironically the more of us that go for badges, the less likely any of us are to be able to progress. My plan is to stick it out here on WW at least until I get to 200m credit in a couple months or until the project runs dry. We’ll see which comes first and how I feel at that point.
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The leading edge is 11,706.982 P (visible on http://www.primegrid.com/server_status_subprojects.php), which means that the project is just over 65% now.

I am confused if this tells us how much the project is finished on an on-going basis.
How, please, can I work out how much is left to be done?
The rate this has picked up recently, WW may not be a viable challenge in September.
(Really I am worried about not getting the next badge)

The leading edge is now 12,612.271P. The limit of the software for project is 18,000P, so that is when the project will end. So we are at 70% now.

For reference, each work unit advances the project by .003P, so my 3070 is advancing the project by about .270P each day.

I like to use the force in these calculations - oh crap - I won't get the next badge.
But I am going to go for it.
It is worthwhile - to try.

Ironically the more of us that go for badges, the less likely any of us are to be able to progress. My plan is to stick it out here on WW at least until I get to 200m credit in a couple months or until the project runs dry. We’ll see which comes first and how I feel at that point.

That is the pragmatic approach.
Unfortunately (and this is great for getting the work done) there are lunatics that want to crunch more than other crunchers.
I put my hand up. I am a lunatic. At least in this.
I have noticed people increasing in WW.

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The leading edge is now 12,612.271P. The limit of the software for project is 18,000P, so that is when the project will end. So we are at 70% now.

For those who want to be very precise, the limit of the software is:

2^64 = 18'446'744'073'709'551'616

so about 18'446.744 P.

Seems like the current 0.003 P work unit size will take us to 18'446.743 P. Then there will be a fraction of a work unit from there to the software limit.

/JeppeSN

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The leading edge is currently 13,385.296P. This means the project is 72.56%. In the last month, the project has averaged 14590 tasks per day, which means the project has about 231 more days before it will be completed at the current rate. There is about 40.49 billion credit left to be claimed for the project.
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Michael Millerick
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The leading edge is currently 13,609.720P. This means the project is 73.78% complete. In the last month, the project has averaged 15660 tasks per day, which means the project has about 205 more days before it will be completed at the current rate. There is about 38.69 billion credit left to be claimed for the project.
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Michael Millerick
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The leading edge is currently 13,799.158P. This means the project is 74.81% complete. In the last month, the project has averaged 16560 tasks per day, which means the project has about 187 more days before it will be completed at the current rate. There is about 37.18 billion credit left to be claimed for the project.
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GDB
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So, the "∞" in the B column for WW on the home page, isn't really true?
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Michael Millerick
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So, the "∞" in the B column for WW on the home page, isn't really true?

It is true from the perspective that the admins do not need to load work for the project and from the perspective that the project could go on forever if the software supported it. Unfortunately the software would need to be rewritten to proceed past a point that we are steadily approaching.
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It is true from the perspective that the admins do not need to load work for the project and from the perspective that the project could go on forever if the software supported it. Unfortunately the software would need to be rewritten to proceed past a point that we are steadily approaching.

Also, right now we are checking all primes between 2^63 and 2^64, i.e. all 64-bit primes, to see if any has the Wieferich property or the Wall-Sun-Sun property. If we were to check 65-bit primes (i.e. all the primes up to 2^65), then it would be slower because these numbers do not seem to fit as well into the hardware. But as you say, we do not even have appropriate software for it.

There are 425656284035217743 primes with at most 64 bits There are 412246861431389469 primes with exactly 65 bits

So a hypothetical project to go from where WW ends to the next power of two would test almost as many prime numbers as we did in total up to the current software limit. However, the expected chance to find something is smaller for bigger primes, so it would still be very lucky if we found a true Wieferich or Wall-Sun-Sun prime (A=0) in such an interval.

/JeppeSN

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Yves Gallot
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For Wall-Sun-Sun prime, it is time to hand over the problem to mathematicians.
No prime for p < 2⁶⁴... probably an hidden property that could be found: no Wall-Sun-Sun prime exists.

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I think Wall-Sun-Sun primes exist. /JeppeSN

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Michael Millerick
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The leading edge is currently 13,966.825P. This means the project is 75.71% complete. In the last month, the project has averaged 16790 tasks per day, which means the project has about 178 more days before it will be completed at the current rate. There is about 35.84 billion credit left to be claimed for the project.
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Michael Millerick
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The leading edge is currently 14,149.069P. This means the project is 76.70% complete. In the last month, the project has averaged 17160 tasks per day, which means the project has about 167 more days before it will be completed at the current rate. There is about 34.38 billion credit left to be claimed for the project.
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