Smallest prime of the form a^2^m + b^2^m, m>=14

[JeppeSN]

I found one for 2^14 and now have:

(3^(2^0)+1^(2^0))/2
(3^(2^1)+1^(2^1))/2
(3^(2^2)+1^(2^2))/2
(5^(2^3)+3^(2^3))/2
(3^(2^4)+1^(2^4))/2
(3^(2^5)+1^(2^5))/2
(3^(2^6)+1^(2^6))/2
(49^(2^7)+9^(2^7))/2
(7^(2^8)+3^(2^8))/2
(35^(2^9)+9^(2^9))/2
(67^(2^10)+57^(2^10))/2
(49^(2^11)+75^(2^11))/2
(157^(2^12)+83^(2^12))/2
(107^(2^13)+69^(2^13))/2
(71^(2^14)+1^(2^14))/2

Would it be interesting to add this sequence, ..., 35, 67, 49, 157, 107, 71, ..., to OEIS?

/JeppeSN

[Batalov]

Yes, why not.
Any sequence is a sequence.
Most of the text can be reused from the previous sequence, and use this sister sequence for the upper boundary https://oeis.org/A275530

[JeppeSN]

Originally Posted by Batalov

Yes, why not.
Any sequence is a sequence.
Most of the text can be reused from the previous sequence, and use this sister sequence for the upper boundary https://oeis.org/A275530

To appear as A301738 (until it is approved, see History to see the proposed versions). /JeppeSN

[JeppeSN]

We can also restrict ourselves to consecutive odd bases:
\[\frac{a^{2^n}+(a-2)^{2^n}}{2}\]
Can also be parametrized in other ways, such as the \(k\) in:
\[\frac{(2k+1)^{2^n}+(2k-1)^{2^n}}{2}\]
OEIS does not seem to have it either (searching for a, or for a-2, or for 2k=a-1, or for k). /JeppeSN

[axn]

Originally Posted by JeppeSN

We can also restrict ourselves to consecutive odd bases:
\[\frac{a^{2^n}+(a-2)^{2^n}}{2}\]
Can also be parametrized in other ways, such as the \(k\) in:
\[\frac{(2k+1)^{2^n}+(2k-1)^{2^n}}{2}\]
OEIS does not seem to have it either (searching for a, or for a-2, or for 2k=a-1, or for k). /JeppeSN

(3^2+1^2)/2
(3^4+1^4)/2
(5^8+3^8)/2
(3^16+1^16)/2
(3^32+1^32)/2
(3^64+1^64)/2
(179^128+177^128)/2
(169^256+167^256)/2
(935^512+933^512)/2
(663^1024+661^1024)/2

[axn]

So, I sieved n=17 for 0 < b < a <= 2048 till 2^53. Not sure what range or what depth Serge has sieved on this n, but it looks "sieved enough". This could be used to divide up PRP work.

Originally Posted by Batalov

Neat!
...Now time for a quick cuda sieve facelift, based on gfnsieve? ^_^

The logic doesn't exactly correspond to gfn/cyclo sieves, so a cuda sieve will have to be built from scratch, more or less; I have to think about it a bit.

[Dr Sardonicus]

Originally Posted by JeppeSN

Would it be interesting to add this sequence, ..., 35, 67, 49, 157, 107, 71, ..., to OEIS?

No. The sequence itself is merely a curiosity. The real interest of this thread, at least for me, has been in the methods described for determining pairs (a, b) for which

a^(2^m) + b^(2^m), [or (a^(2^m) + b^(2^m))/2, if a and b are both odd]

is a (pseudo)prime in a reasonable length of time.

[Batalov]

axn's sieve is easily modified for odd-odd pairs. In fact nearly nothing needs to be changed (only the intake parity filters, if they are there; they were in mine, just drop them).

[axn]

Originally Posted by Batalov

axn's sieve is easily modified for odd-odd pairs. In fact nearly nothing needs to be changed (only the intake parity filters, if they are there; they were in mine, just drop them).

No. The hash matching also needs to change (it currently puts even indexed residues in the hash and uses odd-indexed residues to probe the hash).

[Batalov]

I am researching a very weird anomaly.

From https://oeis.org/A275530 inquiring minds can find out that a(15) > 10,000.
Or in other words, there are no small prps of the GFN' form (a^32768+1)/2.

Now, from what I checked with PFGW, a(15) appears to be either > 160,000 (or even >200,000, which is unreasonably high), or there is a bug in libgwnum. I checked with pfgw, llr, p95 but the speed and results are similar (and the underlying lib is the same). All programs chose the special FFT of size 32K. Changing FFT size to a larger one doesn't help to find a PRP yet. Very strange.

[axn]

Originally Posted by Batalov

I am researching a very weird anomaly.

From https://oeis.org/A275530 inquiring minds can find out that a(15) > 10,000.
Or in other words, there are no small prps of the GFN' form (a^32768+1)/2.

Now, from what I checked with PFGW, a(15) appears to be either > 160,000 (or even >200,000, which is unreasonably high), or there is a bug in libgwnum. I checked with pfgw, llr, p95 but the speed and results are similar (and the underlying lib is the same). All programs chose the special FFT of size 32K. Changing FFT size to a larger one doesn't help to find a PRP yet. Very strange.

Well, the main one (http://oeis.org/A226530) has first three as 70906, 167176, 204462. So > 200k, while low probability, is not _that_ unexpected.