Longest streak of trial-division-detected candidates ever seen by me:
[code]9584 * p(65)#^40 + 1 has factors: 30869
9585 * p(65)#^40 + 1 has factors: 26237
9586 * p(65)#^40 + 1 has factors: 12959
9587 * p(65)#^40 + 1 has factors: 1831
9588 * p(65)#^40 + 1 has factors: 123853
9589 * p(65)#^40 + 1 has factors: 1511
9590 * p(65)#^40 + 1 has factors: 6701
9591 * p(65)#^40 + 1 has factors: 4723
9592 * p(65)#^40 + 1 has factors: 99103
9593 * p(65)#^40 + 1 has factors: 9091
9594 * p(65)#^40 + 1 has factors: 2207[/code]

Sorry, I've been sick... not quite up to my usual posting routine.

[QUOTE=3.14159;226068]Okay: It cannot use vectors and loops, as any method using these is dreadfully inefficient.[/QUOTE]

Sieving (with vectors) = fast, trial division = slow.

[QUOTE=CRGreathouse]Sieving (with vectors) = fast, trial division = slow.
[/QUOTE]

Okay: I need a decent script for that; This is well outside of my range.

I'm unsure I could complete it on my own.

I'll post my recent sieve; maybe you can modify it. It was the work of just a few minutes; I'm sure it could be made much more efficient.

Your sieve is variable-k which is easier to program (but you'll have to make the changes).
[code]\\ Finding terms for A176494
sieve(lim,sz,sm=1)={
my(v=vectorsmall(sz,i,1));
forprime(p=3,lim,
b=znorder(Mod(2,p));
trap(,next,
a=znlog(2293,Mod(2,p),b)
);
if(Mod(2,p)^a!=2293, print("bad at "p);next);
a+=b*ceil((sm-a)/b);
forstep(n=a,sz,b,
v[n]=0
)
);
a=0;
for(i=sm,sz,if(v[i],a++;write("abc.txt", i)));
a
};[/code]

Replace the a=0 and following lines by v (just that one character) if you want to get the vector instead of writing it to a file. This checks for terms with n = sm to n = sz, sieving out the primes up to lim.


Ugh, still sick.

The only ideas I have for code are the beginning, and the end:

I require 5 arguments, not three.

(k value range; n; and primes to sieve range.)

The sieve needs to be generalized towards any arithmetic progression, I guess.

Also: The code doesn't work (Syntax error.)

Well, I'll be off to try and make the appropriate changes.

Blah.. This yields no progress. I'll just give up. Endless failed ideas, with many more soon-to-be failed ideas in mind.

Well, I forfeit.

Now, back to searching for general arithmetic progression primes. The odds will land me something.

(P.S: Found 5358 * 905011[sup]470[/sup] + 1 (≈ 53[sup]2[/sup] digits))

Looking for k * p(125)#^66 + 1 (≈ 19100 digits)

[QUOTE=3.14159;226139]Also: The code doesn't work (Syntax error.)[/QUOTE]

That's interesting. My version says
[code]znlog(x,g,{o}): return the discrete logarithm of x in (Z/nZ)* in base g. If
present, o represents the multiplicative order of g. If no o is given, assume
that g generate (Z/nZ)*.[/code]
but I see that my Windows version says
[code]znlog(x,g): g as output by znprimroot (modulo a prime). Return smallest
non-negative n such that g^n = x.[/code]
so I guess you'll just have to drop that trailing ,b.

The odds of finding something:

Normally: 1 in about 44000.

Potential prime factors eliminated: first 125 primes:
Leaving it at about 1 in 3750. (w/no sieving)

[QUOTE=3.14159;226139]The only ideas I have for code are the beginning, and the end:

I require 5 arguments, not three.

(k value range; n; and primes to sieve range.)[/QUOTE]

Four -- the code should probably just start at 2 or 3 for the prime range. All you need to add in terms of arguments is the n.

[QUOTE=3.14159;226139]The sieve needs to be generalized towards any arithmetic progression, I guess.[/QUOTE]

Changed, not generalized, from variable-n to variable-k.

CRG if you're sick take a rest or Questions will make you worse.