Special primes

This continues my older primes page. I have collected to this page some more special type of primes and polynomials that generates primes.

Zeroes

Primes that are form 10^x+y, where x would be something very big and y either 1, 3, 7 or 9 to make it prime. So for example 10¹⁷+3 = 100 000 000 000 000 003, but much, much more zeroes.

Here are some highest ones:

10⁴⁵⁶² + 9
10⁶³⁴⁴ + 7
10⁷⁶⁶⁸ + 3

Following are not certainly primes but there is high probability for that:

10²⁶⁹²⁷ + 3
10³⁰²²¹ + 7
10³¹⁸¹⁰ + 9

Interestingly I haven't found any similar primes ending to number 1, except 11 and 101. Is there any or is it proven there can't be any (and if so, why)?

Ones

Primes that are form (10^x-1)/9, where x is something very big. This generates numbers that contains only ones. For example (10¹⁹-1)/9 = 1 111 111 111 111 111 111, which is prime. Obviously numbers with only ones can be primes, number containing only threes could be divided by 3 etc.

These seem to be much harder to find than "zeroes" type above. Here are all 6 I've found:

(10² - 1) / 9
(10¹⁹ - 1) / 9
(10²³ - 1) / 9
(10³¹⁷ - 1) / 9
(10¹⁰³¹ - 1) / 9
(10⁴⁹⁰⁸¹ - 1) / 9   (not certainly prime but there is high probability for that)

2, 19, 23, 317, 1031, 49081. Is there any logic in those? Other than that they are also primes :-)

In case someone is wondering, I was running my program for all possible values up to that 49081. No other than those 6 were even probable prime. I later found this page about repunit primes. See that page for even bigger similar probable primes found.

One multipliers

Polynomials where all multipliers are 1. Unfortunately these do not work with n=0, since 1 itself isn't prime.

n⁷ + n⁴ + n² + n + 1   n=[1..5]
n¹¹ + n¹⁰ + n⁹ + n⁸ + n⁷ + n⁵ + 1   n=[1..7]
n¹⁶ + n¹⁵ + n¹⁰ + n⁹ + n⁸ + n⁶ + n⁵ + n³ + n² + n + 1   n=[1..8]
n²⁵ + n²² + n²¹ + n²⁰ + n¹⁹ + n¹⁶ + n¹⁵ + n¹³ + n¹² + n¹¹ + n¹⁰ + n⁸ + n⁷ + n⁵ + n⁴ + n + 1   n=[1..9]

Big results

Usually any prime generating polynomial generates rather small primes. This is expected since small numbers are much more likely to be primes than big ones. Here is some polynomials with big exponents and results are getting huge really fast.

n⁵⁸ + n⁵⁰ + 101   n=[0..3]
n⁸¹ + n⁵⁸ + 857   n=[0..4]
n¹⁵² + n¹⁰⁰ + 11   n=[0..4]
n³⁶⁷ + n³⁶² + 17   n=[0..4]

From last one, with value n=4, result is: 90 456 697 114 872 591 699 151 142 601 866 986 247 973 054 988 995 842 587 515 218 097 945 661 753 757 139 450 110 024 031 878 460 840 474 072 254 241 909 265 393 011 852 735 167 210 041 163 028 615 597 492 471 611 437 305 305 750 737 092 121 072 031 909 872 401 602 920 806 418 550 777 446 417

Repeating

Few ones where I've bump into while playing around with these. These makes lots of primes but same ones are repeating.

n⁴ - 94*n³ + 2999*n² - 37130*n + 153313   n=[0..47] (22 distinct)
n⁴ - 102*n³ + 4502*n² - 96951*n + 733517   n=[0..51] (26 distinct)
n⁴ - 110*n³ + 4841*n² - 99880*n + 806371   n=[0..55] (28 distinct)
n² - 79*n + 1601   n=[0..79] (40 distinct)

Links

One of my favorite pages with primes: Prime-Generating Polynomial

Another great page, this is for testing primality of huge numbers: Factorization using the Elliptic Curve Method

Simple and quick JavaScript test: Prime Curios!: A Primality Test