Sieve needed for a^(2^b)+(a+1)^(2^b)

[robert44444uk]

Originally Posted by R.D. Silverman

Question. Singular.

Q1: A more interesting question is whether there exists values of a for
which there are no primes.

Q2: If so, what is the density of that set?

That is a prime number of questions.

Originally Posted by R.D. Silverman

The distribution of primes is an interesting topic of research.

Ok, but why isn't the density of primes, a variable and b fixed, not of equal interest? For series with larger b the absolute size of values in the series grows very quickly and its prime probability decreases rapidly.

For b=0, It is possible to demonstrate that the number of primes in the series is infinite. Values are 2a+1, and Dirichlet's AP theory nails that.

A question is whether other values b produce an infinite number of primes. Open question, I would imagine.

One can imagine that at some value of b, there is more than the chance that the number of primes becomes finite, and that, for very large b there are values that do not produce primes - possibly not provable.

[R.D. Silverman]

Originally Posted by robert44444uk

Q1: A more interesting question is whether there exists values of a for
which there are no primes.

Q2: If so, what is the density of that set?

That is a prime number of questions.



Ok, but why isn't the density of primes, a variable and b fixed, not of equal interest? .

Because it is ALREADY a known, open problem. Look up
Schinzel's Conjecture and the Bateman-Horn conjecture.

The exact answer to this latter question is well known, but a formal
proof is lacking (for much the same reason as we have no formal proof
of the twin-prime conjecture).

[robert44444uk]

Originally Posted by R.D. Silverman

Because it is ALREADY a known, open problem. Look up
Schinzel's Conjecture and the Bateman-Horn conjecture.

The exact answer to this latter question is well known, but a formal
proof is lacking (for much the same reason as we have no formal proof
of the twin-prime conjecture).

Mmmm, very interesting. Thank you Bob

[robert44444uk]

Geoff

What is the maximum value that a can take in your program? I was hoping to look at exponent b=10 and I need to start a at 10^10 in order to get 10000 digit prps. But it is telling me I am out of range.

[robert44444uk]

I would like to reserve b=11, i.e. exponent is 2^11=2048. I have already sieved quite deeply and have already found 16 prps that are 10000 digits or more, and these have been submitted to the Lifchitz site.

The first few a's (smaller than 10000 digits) are:

a     digits
754	5894
1289	6371
1368	6424
3159	7168
3280	7201
3301	7207
4976	7572
6204	7768
6283	7780
6723	7839
6904	7863
7141	7893
10246	8214
11417	8311
13268	8444

[geoff]

Originally Posted by robert44444uk

Geoff

What is the maximum value that a can take in your program? I was hoping to look at exponent b=10 and I need to start a at 10^10 in order to get 10000 digit prps. But it is telling me I am out of range.

a is limited to about 2^31. I could increase that to 2^62 if necessary.

[robert44444uk]

Originally Posted by geoff

a is limited to about 2^31. I could increase that to 2^62 if necessary.

That would be great Geoff.

I tested b=11 up to a=100000 and I am well on the way to a=200000.

Results for first 100000:

Average prp size = 9250 digits. Digits at a=50000, 9624

Expected number of prps in population of 100000 odd numbers at that digits length:
100000/2*ln(10^9250)= 9.38; 100000/2*(ln(10^9624)= 9.02

prps found: 79

[robert44444uk]

Results up to a=200000, 72 of which are of interest to the Lifchitz prp site.

prp, no of digits
754^2048+755^2048,5895
1289^2048+1290^2048,6371
1368^2048+1369^2048,6424
3159^2048+3160^2048,7168
3280^2048+3281^2048,7201
3301^2048+3302^2048,7207
4976^2048+4977^2048,7572
6204^2048+6205^2048,7768
6283^2048+6284^2048,7780
6723^2048+6724^2048,7839
6904^2048+6905^2048,7863
7141^2048+7142^2048,7893
10246^2048+10247^2048,8214
11417^2048+11418^2048,8311
13268^2048+13269^2048,8444
15456^2048+15457^2048,8580
19428^2048+19429^2048,8784
19683^2048+19684^2048,8795
19698^2048+19699^2048,8796
20298^2048+20299^2048,8822
21484^2048+21485^2048,8873
22543^2048+22544^2048,8916
23702^2048+23703^2048,8960
23815^2048+23816^2048,8965
24747^2048+24748^2048,8999
27010^2048+27011^2048,9077
32319^2048+32320^2048,9236
34133^2048+34134^2048,9285
36201^2048+36202^2048,9337
37030^2048+37031^2048,9357
39438^2048+39439^2048,9413
41292^2048+41293^2048,9454
44472^2048+44473^2048,9520
47623^2048+47624^2048,9580
50198^2048+50199^2048,9628
51031^2048+51032^2048,9642
51370^2048+51371^2048,9648
51521^2048+51522^2048,9651
52628^2048+52629^2048,9670
53073^2048+53074^2048,9677
53309^2048+53310^2048,9681
53767^2048+53768^2048,9689
55911^2048+55912^2048,9724
56630^2048+56631^2048,9735
59424^2048+59425^2048,9778
59583^2048+59584^2048,9780
61797^2048+61798^2048,9813
61930^2048+61931^2048,9815
63346^2048+63347^2048,9835
64551^2048+64552^2048,9851
66076^2048+66077^2048,9872
66396^2048+66397^2048,9877
66674^2048+66675^2048,9880
67392^2048+67393^2048,9890
67704^2048+67705^2048,9894
68934^2048+68935^2048,9910
69349^2048+69350^2048,9915
69521^2048+69522^2048,9917
70510^2048+70511^2048,9930
70583^2048+70584^2048,9931
70601^2048+70602^2048,9931
71168^2048+71169^2048,9938
72189^2048+72190^2048,9951
75127^2048+75128^2048,9986
76974^2048+76975^2048,10008
77074^2048+77075^2048,10009
78493^2048+78494^2048,10025
81114^2048+81115^2048,10055
81544^2048+81545^2048,10059
83794^2048+83795^2048,10084
83896^2048+83897^2048,10085
88373^2048+88374^2048,10131
88830^2048+88831^2048,10135
89768^2048+89769^2048,10145
91595^2048+91596^2048,10163
93534^2048+93535^2048,10181
93782^2048+93783^2048,10184
94243^2048+94244^2048,10188
95891^2048+95892^2048,10202
100757^2048+100758^2048,10248
101197^2048+101198^2048,10251
103577^2048+103578^2048,10272
112746^2048+112747^2048,10348
114562^2048+114563^2048,10362
115780^2048+115781^2048,10371
119216^2048+119217^2048,10396
119791^2048+119792^2048,10401
120264^2048+120265^2048,10405
121334^2048+121335^2048,10413
122038^2048+122039^2048,10418
122543^2048+122544^2048,10422
122927^2048+122928^2048,10424
127699^2048+127700^2048,10458
132663^2048+132664^2048,10492
134315^2048+134316^2048,10503
135212^2048+135213^2048,10509
136141^2048+136142^2048,10515
136907^2048+136908^2048,10520
139844^2048+139845^2048,10539
141774^2048+141775^2048,10551
143390^2048+143391^2048,10561
143774^2048+143775^2048,10564
143878^2048+143879^2048,10564
147617^2048+147618^2048,10587
147696^2048+147697^2048,10588
149603^2048+149604^2048,10599
149622^2048+149623^2048,10599
150635^2048+150636^2048,10605
152562^2048+152563^2048,10617
153794^2048+153795^2048,10624
154919^2048+154920^2048,10630
155382^2048+155383^2048,10633
156064^2048+156065^2048,10637
158330^2048+158331^2048,10650
159409^2048+159410^2048,10656
159450^2048+159451^2048,10656
160497^2048+160498^2048,10662
163585^2048+163586^2048,10679
164367^2048+164368^2048,10683
164376^2048+164377^2048,10683
166566^2048+166567^2048,10695
168247^2048+168248^2048,10704
168501^2048+168502^2048,10705
169670^2048+169671^2048,10711
174775^2048+174776^2048,10737
179939^2048+179940^2048,10763
181393^2048+181394^2048,10769
182116^2048+182117^2048,10774
182749^2048+182750^2048,10777
183451^2048+183452^2048,10780
184125^2048+184126^2048,10784
184607^2048+184608^2048,10786
185491^2048+185492^2048,10790
186829^2048+186830^2048,10797
194559^2048+194560^2048,10833
195325^2048+195326^2048,10836

[geoff]

I have uploaded a new version of the sieve 0.2 that allows a up to 2^62-2. It will use a bit more memory and might be a bit slower than the previous version. I haven't tested it at all, so let me know if there are any problems.

[robert44444uk]

Originally Posted by henryzz

i accidentally left b=12 running quite a lot longer than i meant to and found:
311^(2^12)+312^(2^12)
2741^(2^12)+2742^(2^12)
3582^(2^12)+3583^(2^12)
5293^(2^12)+5294^(2^12)
are all 3-prp
i expect they are already known

Henryzz, are you still running this or is this 2^12 now free? What was the highest value you checked? I might carry on from where you left off if you are done.

[henryzz]

Originally Posted by robert44444uk

Henryzz, are you still running this or is this 2^12 now free? What was the highest value you checked? I might carry on from where you left off if you are done.

i only ran this for a couple a days
i only meant to run for a few hours as a test but i forgot to stop it
i have not a clue where i searched up to