A fun start for the new year: searching among somewhat larger PRPs, I spotted the Wagstaff Prime [URL="http://factorization.ath.cx/index.php?id=1100000000004299235"](2^5807+1)/3[/URL]. Wagstaff primes are especially attractive for these proofs because N+1 and N-1 are Cunningham numbers with different exponents, increasing the likelihood that one of them has lots of algebraic factors. Like most of the proofs I feature in this thread, this one was missing algebraic factors from one side. This hints that a systematic check of Wagstaff Primes may turn up more easily completed proofs in the factordb. The exponents for Wagstaff Primes are [URL="http://oeis.org/A000978"]OEIS Sequence 978[/URL]

I found another of these (2^n +/- a)/b PRPs where a+b or a-b is a power of 2, leading to sufficient algebraic factors for an N+1 or N-1 proof. Today's example was from near the clearing edge of PRPs: [URL="http://factorization.ath.cx/index.php?id=1100000000207996048"](2^4601+7)/39[/URL]

I guess I've not inspired others to browse the PRP list for easy targets. I found this one among the 50 smallest PRPs.

[URL="http://factorization.ath.cx/index.php?id=1100000000031632945"](10^1410*74-11)/63[/URL] - the N-1 cancels the 74, leaving lots of algebraic factors for 10^1410-1.

[URL="http://www.factordb.com/index.php?query=%2810%5E3258-73%29%2F9"](10^3258-73)/9[/URL] via a p1057 and some algebra in N+1

...and [URL="http://www.factordb.com/index.php?query=%2810%5E12891%2B11%29%2F3"](10^12891+11)/3[/URL] with N-1 (these are some old toys)

[URL="http://www.factordb.com/index.php?id=1100000000482470622"]2*911[sup]381[/sup]+1[/URL] via N-1. It's the only prime of the form 2*911[sup]n[/sup]+1 that I've found, other than 1823. I used the factor tables. Here's the link to the table (Near Cunningham):

[URL]http://www.factordb.com/index.php?query=k*b%5En%2Bd&use=n&k=2&b=911&n=1&d=1&VP=on&VC=on&EV=on&OD=on&PR=on&PRP=on&U=on&perpage=20&format=1&sent=Show[/URL]

[QUOTE=Stargate38;286099]It's the only prime of the for 2*911[sup]n[/sup]+1 that I've found, other than 1823. I used the factor tables.[/QUOTE]

Are you aware that factor tables is an inefficient search method and has the unfortunate side effect of queueing ALL of these numbers to be fully factored?

[URL="http://www.factordb.com/index.php?id=1100000000482633606"]2*911[sup]2171[/sup]+1[/URL] is prime (N-1).

Proth is faster. I just plug the prime numbers I find into the db when I find them.

[URL="http://factordb.com/index.php?id=1100000000294641741"]3^8890-2[/URL] needed the [URL="http://factordb.com/index.php?id=1100000000484273807"]p1402[/URL] from [URL="http://factordb.com/index.php?id=1000000000047128914"]3^8890-1[/URL] proven to complete an N+1 proof.

Although I don't know why there is interest in (3^3333+4), I enjoy finding cases like [URL="http://factorization.ath.cx/index.php?id=1100000000208024795"](3^3333+4)/31[/URL] in the PRP lists, where I can visually spot that N-1 has a difference of powers of 3 (3^3333-3^3) that leaves the cyclotomic number (3^3330-1). In many cases, including this one, helping the factordb find the algebraic factors of the cyclotomic number completes the primality proof.

Might be worth adding all the algebraic factorizations for cyclotomic numbers.

[QUOTE=henryzz;287450]Might be worth adding all the algebraic factorizations for cyclotomic numbers.[/QUOTE]

It's more complicated than that. The factordb already finds most of the algebraic factors for pure cyclotomic numbers. But it isn't capable of finding hidden cyclotomic numbers. This case was factoring (3^3333+4)/31-1. It would require a clever factoring process to automatically see this is (3^3330-1)*27/31, and therefore all the factors of 3^3330-1 except for 31 divide the number.