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Old 2019-05-03, 23:06   #1
Godzilla
 
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May 2016

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Default help about ‘function’ that generates prime numbers

Good evening ,

I need to understand if this could be a valid function to generate prime numbers and or if it is already known.


I wrote a code in python language, I hope it is more understandable than the others because I put the comments.

in practice the function returns a prime number or composite but the peculiarity of this function is that many composite numbers if they are divided by the number 11 return a prime number.

for example up to 100 the function returns 24 prime numbers then if I divide the composite numbers with the number 11 I find another 9 prime numbers, in total 33 prime numbers.

another example up to 1000 the function returns 156 prime numbers then if I divide the composite numbers by the number 11 I find another 91 prime numbers.

important note all the prime numbers are different from each other.


My question is did I find something interesting or not?

The Function is :

Function = n^2-1+n^2-1-n^2-1+n^2-1+n^2

n= odd number

Thanks to those who will respond.
Good weekend ,

Alessandro Boatto

ProgFunction.py


Code:
AA=0 # count-prime-found function 
BB=0 # count-prime-found function f/11
CC=1 # count-prime-found ( Is 1 because the number 2 is prime and the program start from the number 3)
c = 3 # N
bb=0 # start input
I=1 # while function 
II=1 # while function f/11
III=1 # while N  
b=0 # function
d=0 # function f/11
print('The program Test the prime number for :\n\nFunction = n^2-1+n^2-1-n^2-1+n^2-1+n^2 \n\n and\n\nFunction/11\n\n and\n\nN\n\n');
bb=input('Insert number example 100 : ')

while c<=bb:

    
    b=c**2-1+c**2-1-c**2-1+c**2-1+c**2
    d=b/11.0
    print(' N= ---->' , c)


    while I<=b :


        I=I*1
        I=I+1
        if b == I:

         print(' PRIME NUMBER -----', b)
         
         AA=AA+1
         I=1
         break
         
         
        if b%I==0 :

    



         print('NO PRIME--->',b)
         I=1
         break
         print('f/11=',d)
    while II<=d :


        II=II*1
        II=II+1
        if d == II:

         print(' f/11 PRIME NUMBER -----', d)
         
         BB=BB+1
         II=1
         break
         
         
        if d%II==0 :

    



         print('NO PRIME--->',d)
         II=1
         break

    while III<=c :


        III=III*1
        III=III+1
        if c == III:

         print(' N PRIME NUMBER -----', c)
         
         CC=CC+1
         III=1
         break
         
         
        if c%III==0 :

    



         print('N NO PRIME--->',c)
         III=1
         break
    c=c+2
print('TOTAL PRIME NUMBER Function FOUND : ',AA)
print('TOTAL PRIME NUMBER f/11 FOUND : ',BB)
print('TOTAL function + f/11 :  ',AA+BB)
print('\n\n\n')
print('TOTAL PRIME NUMBER N FOUND : ',CC)
print('TOTAL function + f/11 + N : ',AA+BB+CC)

OUTPUT

Code:
The program Test the prime number for :

Function = n^2-1+n^2-1-n^2-1+n^2-1+n^2 

 and

Function/11

 and

N


Insert number example 100 : 100
(' N= ---->', 3)
(' PRIME NUMBER -----', 23)
(' N PRIME NUMBER -----', 3)
(' N= ---->', 5)
(' PRIME NUMBER -----', 71)
(' N PRIME NUMBER -----', 5)
(' N= ---->', 7)
('NO PRIME--->', 143)
(' f/11 PRIME NUMBER -----', 13.0)
(' N PRIME NUMBER -----', 7)
(' N= ---->', 9)
(' PRIME NUMBER -----', 239)
('N NO PRIME--->', 9)
(' N= ---->', 11)
(' PRIME NUMBER -----', 359)
(' N PRIME NUMBER -----', 11)
(' N= ---->', 13)
(' PRIME NUMBER -----', 503)
(' N PRIME NUMBER -----', 13)
(' N= ---->', 15)
('NO PRIME--->', 671)
(' f/11 PRIME NUMBER -----', 61.0)
('N NO PRIME--->', 15)
(' N= ---->', 17)
(' PRIME NUMBER -----', 863)
(' N PRIME NUMBER -----', 17)
(' N= ---->', 19)
('NO PRIME--->', 1079)
(' N PRIME NUMBER -----', 19)
(' N= ---->', 21)
(' PRIME NUMBER -----', 1319)
('N NO PRIME--->', 21)
(' N= ---->', 23)
(' PRIME NUMBER -----', 1583)
(' N PRIME NUMBER -----', 23)
(' N= ---->', 25)
(' PRIME NUMBER -----', 1871)
('N NO PRIME--->', 25)
(' N= ---->', 27)
('NO PRIME--->', 2183)
('N NO PRIME--->', 27)
(' N= ---->', 29)
('NO PRIME--->', 2519)
(' f/11 PRIME NUMBER -----', 229.0)
(' N PRIME NUMBER -----', 29)
(' N= ---->', 31)
(' PRIME NUMBER -----', 2879)
(' N PRIME NUMBER -----', 31)
(' N= ---->', 33)
('NO PRIME--->', 3263)
('N NO PRIME--->', 33)
(' N= ---->', 35)
(' PRIME NUMBER -----', 3671)
('N NO PRIME--->', 35)
(' N= ---->', 37)
('NO PRIME--->', 4103)
(' f/11 PRIME NUMBER -----', 373.0)
(' N PRIME NUMBER -----', 37)
(' N= ---->', 39)
('NO PRIME--->', 4559)
('N NO PRIME--->', 39)
(' N= ---->', 41)
(' PRIME NUMBER -----', 5039)
(' N PRIME NUMBER -----', 41)
(' N= ---->', 43)
('NO PRIME--->', 5543)
(' N PRIME NUMBER -----', 43)
(' N= ---->', 45)
('NO PRIME--->', 6071)
('N NO PRIME--->', 45)
(' N= ---->', 47)
('NO PRIME--->', 6623)
(' N PRIME NUMBER -----', 47)
(' N= ---->', 49)
('NO PRIME--->', 7199)
('N NO PRIME--->', 49)
(' N= ---->', 51)
('NO PRIME--->', 7799)
(' f/11 PRIME NUMBER -----', 709.0)
('N NO PRIME--->', 51)
(' N= ---->', 53)
(' PRIME NUMBER -----', 8423)
(' N PRIME NUMBER -----', 53)
(' N= ---->', 55)
('NO PRIME--->', 9071)
('N NO PRIME--->', 55)
(' N= ---->', 57)
(' PRIME NUMBER -----', 9743)
('N NO PRIME--->', 57)
(' N= ---->', 59)
('NO PRIME--->', 10439)
(' f/11 PRIME NUMBER -----', 949.0)
(' N PRIME NUMBER -----', 59)
(' N= ---->', 61)
(' PRIME NUMBER -----', 11159)
(' N PRIME NUMBER -----', 61)
(' N= ---->', 63)
(' PRIME NUMBER -----', 11903)
('N NO PRIME--->', 63)
(' N= ---->', 65)
(' PRIME NUMBER -----', 12671)
('N NO PRIME--->', 65)
(' N= ---->', 67)
(' PRIME NUMBER -----', 13463)
(' N PRIME NUMBER -----', 67)
(' N= ---->', 69)
('NO PRIME--->', 14279)
('N NO PRIME--->', 69)
(' N= ---->', 71)
('NO PRIME--->', 15119)
(' N PRIME NUMBER -----', 71)
(' N= ---->', 73)
('NO PRIME--->', 15983)
(' f/11 PRIME NUMBER -----', 1453.0)
(' N PRIME NUMBER -----', 73)
(' N= ---->', 75)
(' PRIME NUMBER -----', 16871)
('N NO PRIME--->', 75)
(' N= ---->', 77)
(' PRIME NUMBER -----', 17783)
('N NO PRIME--->', 77)
(' N= ---->', 79)
(' PRIME NUMBER -----', 18719)
(' N PRIME NUMBER -----', 79)
(' N= ---->', 81)
('NO PRIME--->', 19679)
(' f/11 PRIME NUMBER -----', 1789.0)
('N NO PRIME--->', 81)
(' N= ---->', 83)
(' PRIME NUMBER -----', 20663)
(' N PRIME NUMBER -----', 83)
(' N= ---->', 85)
('NO PRIME--->', 21671)
('N NO PRIME--->', 85)
(' N= ---->', 87)
('NO PRIME--->', 22703)
('N NO PRIME--->', 87)
(' N= ---->', 89)
('NO PRIME--->', 23759)
(' N PRIME NUMBER -----', 89)
(' N= ---->', 91)
('NO PRIME--->', 24839)
('N NO PRIME--->', 91)
(' N= ---->', 93)
(' PRIME NUMBER -----', 25943)
('N NO PRIME--->', 93)
(' N= ---->', 95)
('NO PRIME--->', 27071)
(' f/11 PRIME NUMBER -----', 2461.0)
('N NO PRIME--->', 95)
(' N= ---->', 97)
('NO PRIME--->', 28223)
(' N PRIME NUMBER -----', 97)
(' N= ---->', 99)
(' PRIME NUMBER -----', 29399)
('N NO PRIME--->', 99)
('TOTAL PRIME NUMBER Function FOUND : ', 24)
('TOTAL PRIME NUMBER f/11 FOUND : ', 9)
('TOTAL function + f/11 :  ', 33)




('TOTAL PRIME NUMBER N FOUND : ', 25)
('TOTAL function + f/11 + N : ', 58)
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Old 2019-05-05, 12:58   #2
Godzilla
 
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Updating

Up to 10000

Function returns 1193 prime numbers
Function /11 returns 909 prime numbers

Function + Function/11 = 2102 prime numbers

and continuing the values of the two functions are equivalent ... these results could be interesting or not? in practice the number 11 is very often present if it is divided by it, and why?
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Old 2019-05-05, 14:00   #3
kar_bon
 
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Frst:
Why such cryptic function?
What is "a-1+a-1-a-1+a-1+a"?

Second:
- download a version of PFGW for your system
- create a text-file "test.txt" with
Code:
ABC2 3*$a^2-4
a: from 1 to 10000
- run this by calling "pfgw -l test.txt"
- got a result in seconds, see the created file "pfgw-prime.log"
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Old 2019-05-05, 15:27   #4
Godzilla
 
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Quote:
Originally Posted by kar_bon View Post
Frst:
Why such cryptic function?
What is "a-1+a-1-a-1+a-1+a"?

Second:
- download a version of PFGW for your system
- create a text-file "test.txt" with
Code:
ABC2 3*$a^2-4
a: from 1 to 10000
- run this by calling "pfgw -l test.txt"
- got a result in seconds, see the created file "pfgw-prime.log"

thank you for simplifying the function. I have a Mac 64 and I downloaded the PFGW software, I put the folder on the desktop pfgw_mac_3 and I created the file test.txt inside with the code ABC2 ... and I opened the file PFGW64 and the output is this after 6 times that I pressed continue:

Code:
Last login: Sun May  5 17:07:42 on ttys007
/Users/alessandroboatto/Desktop/pfgw_mac_3/pfgw64 ; exit;
AirdiAlessandro:~ alessandroboatto$ /Users/alessandroboatto/Desktop/pfgw_mac_3/pfgw64 ; exit;

--===COPYRIGHT AND LICENSE===--

PrimeForm/GW - a program to perform a variety of primality tests.
Copyright (C) 1999-2011, The OpenPFGW project at sourceforge.

See the accompanying LICENSE.pfgw for the Terms and Conditions
regarding the use of this product and third-party libraries therein.

This product uses the gwnum libraries (version 25) by George Woltman.
Copyright 1995-2009 Mersenne Research, Inc., all rights reserved.
See the accomanying LICENSE file, (also see http://www.mersenne.org/prize.htm)

This product uses the GNU Multiple Precision Library (version 5.0.1).
Copyright (C) 1991-2011 Free Software Foundation, Inc.
See the accompanying COPYING.LIB for Terms and Conditions.

 -- Press enter to continue, or ^C to break --
--===BASIC USAGE===--

pfgw supports a variety of command-line options. The simplest is

pfgw filename

which proceeds directly to a probable primality test of every number in the
specified file, one expression per line.

--===OPTIONS===--

In addition, the following command-line flags are supported. Flags may appear
in any order on the command-line, and are prefixed by a '-' sign.

-q<expression> enter the "q"uick expression to be tested.

--  Use a double dash to have PFGW query you for an expression to test Any
file name entered will be ignored when using either the -- or -q options

-l<logfilename> will output all text not only to the screen, but also to 
the logfilename file (if <logfilename> is not specified, then pfgw.out is used

-- Press enter to continue, or ^C to break --
--===MORE OPTIONS===--

-? This help message.

-f<percent> trial-factors every number before attempting any primality test. If
a factor is found, the PRP test is not performed. Optional <percent> is the
percent of what pfgw "normally" trial factors to.  i.e. default is -f100.
The <percent> can be from 1 up to 65535.  <percent> ignored if -e switch used.

-d will perform deep factoring, when used along with -f. In -f mode, if a
factor is found, processing of that number stops. The -d flag continues to
find factors until the trial-factoring limit, or until the number is
completely factored. If a cofactor exists, it is passed to the primality
test.

-s<number>
-e<number>
Used in conjunction with the -f and -v flags, the minimum and maximum
values for trial-factoring can be set. The defaults are -s0, and -e
determined by an internal algorithm.

-k produces terse output

-o (or -od) factor [O]nly mode. In this mode instead of performing a primality
test, the expression or factors are written to stdout.  If the switch -od is
used, then the decimal expansion of the number is output.
-- Press enter to continue, or ^C to break --
--===MORE OPTIONS===--

-b<number> allows you to override the PRP testing base, which defaults
to 3. Legal bases are in the range 2 to 255.

-v will process the numbers in the input file in modular vector form.
For each prime in a factor range set by the -s and -e flags, the value
of each number modulo that prime is output to stdout.

-g<o> Test for factors of GFN's. If -go then ONLY test for factors (no PRP).
-gx<o> Test for factors of "extended" GFN's. xGFN's form is a^2^n+b^2^n

-- Press enter to continue, or ^C to break --
--===ADVANCED OPTIONS FOR DETERMINSTIC TESTS===--

-t currently performs a deterministic test. By default this is an N-1
test, but N+1 testing may be selected with '-tp'. N-1 or N+1 is
factored, and Pocklington's or Morrison's Theorem is applied. If 33%
size of N prime factors are available, the Brillhart-Lehmer-Selfridge
test is applied for conclusive proof of primality. If less than 33%
is factored, this test provides 'F-strong' probable primality with
respect to the factored part F.

-tc performs a combined N-1 and N+1 test. Both tests are executed to
get strong probable primality to two factored parts F and G.
If F<G and F.G^3>N, the N+1 variant of BLS can give a proof.
If F>G and F^3.G>N, the N-1 variant of BLS can give a proof.
In either case, results just short of the target may still give a
proof with sufficient loop tests for perfect squares.

-h<filename> allows you to include known factors of N-1, N, or N+1
from a file. It is your responsibility to ensure this file includes
only prime expressions, one per line, or the results may be incorrect.
This is most often useful to provide known factors discovered using
another program, which will be sufficient to complete a proof.

-- Press enter to continue, or ^C to break --
--===ADVANCED OPTIONS FOR DETERMINSTIC TESTS===--

-x<depth> for deterministic tests to prove a number prime, factorization of
33% of N-1 (or N+1) is required.  Ocasionally, a number will be just short
of this required amount. If you receive a message stating "Proof incomplete
rerun with -x24000" then rerun the proof using the -x switch with whatever
number pfgw stated.  PFGW will attempt to perform some square free loops to
attempt to prove the number (even though it is just short of 33% factorized)

-r enables round off error checking for all iterations of all tests

-a<number> Authenticates composites.  Using a -a0 will PRP test using the
"normal" FFT sizes.  A switch of -a1 will test using 1 less bit per
FFT limb.  This will take more time but will also eliminate any chance of
rounding errors inherant with FFT multiplies.  The lower 62 bits of the
results of a PRP test will be listed (on composites only).  This allows
checking for math library errors.  If a run on the same numbers using -a0
and a -a1 results in different residues, then there were round off errors.
-- Press enter to continue, or ^C to break --
logout
Saving session...
...copying shared history...
...saving history...truncating history files...
...completed.

[Processo completato]

I tried to give the command from the terminal that you gave me (also moving to the pfgw_mac_3 folder), pfgw -l test.txt but the computer can't find the pfgw command, where am I doing wrong?
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Old 2019-05-05, 16:18   #5
a1call
 
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Do you have a Windows computer that you can use?
There are likely more people who know how to use Windows than Mac. Including me which might be able to help
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Old 2019-05-06, 06:47   #6
kar_bon
 
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Can't help either, no MAC.
But: You tried in command line "pfgw ...", why? Try "pfgw64 ..."?
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Old 2019-05-06, 07:15   #7
M344587487
 
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Try opening a terminal, cd to the pfgw directory then run the executable with ./pfgw64. That'll have the same effect as running the executable directly as you did, but you can now add the commandline options.

Last fiddled with by M344587487 on 2019-05-06 at 07:18 Reason: Clarity
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Old 2019-05-09, 13:55   #8
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Thanks now works with the ./pfgw64 command .

And about this function has anyone seen something similar ? and about that it is very often divisible by the number 11?
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Old 2019-05-10, 19:48   #9
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Quote:
Originally Posted by Godzilla View Post

and about that it is very often divisible by the number 11?
I discovered that with this function if I divide by the number 11 always and only every 8 or 14 sequences always generates a prime number.

Example looks the output :

for example if you look at the output of the function N = 7 it generates the number 143 which is not prime but if I divide it by the number 11 generas the prime number 13, then the function N = 15 generates the number 671 which is not a prime number but if I divide it by the number 11 it generates the number 61 which is a prime number and eN = 15 is 8 units from the number N = 7, then the number N = 29 is 14 units from the number N = 15 ....

Code:
Insert number example 100 : 100
(' N= ---->', 3)
(' PRIME NUMBER -----', 23)
(' N PRIME NUMBER -----', 3)
(' N= ---->', 5)
(' PRIME NUMBER -----', 71)
(' N PRIME NUMBER -----', 5)
(' N= ---->', 7)
('NO PRIME--->', 143)
(' f/11 PRIME NUMBER -----', 13.0)
(' N PRIME NUMBER -----', 7)
(' N= ---->', 9)
(' PRIME NUMBER -----', 239)
('N NO PRIME--->', 9)
(' N= ---->', 11)
(' PRIME NUMBER -----', 359)
(' N PRIME NUMBER -----', 11)
(' N= ---->', 13)
(' PRIME NUMBER -----', 503)
(' N PRIME NUMBER -----', 13)
(' N= ---->', 15)
('NO PRIME--->', 671)
(' f/11 PRIME NUMBER -----', 61.0)
('N NO PRIME--->', 15)
(' N= ---->', 17)
(' PRIME NUMBER -----', 863)
(' N PRIME NUMBER -----', 17)
(' N= ---->', 19)
('NO PRIME--->', 1079)
(' N PRIME NUMBER -----', 19)
(' N= ---->', 21)
(' PRIME NUMBER -----', 1319)
('N NO PRIME--->', 21)
(' N= ---->', 23)
(' PRIME NUMBER -----', 1583)
(' N PRIME NUMBER -----', 23)
(' N= ---->', 25)
(' PRIME NUMBER -----', 1871)
('N NO PRIME--->', 25)
(' N= ---->', 27)
('NO PRIME--->', 2183)
('N NO PRIME--->', 27)
(' N= ---->', 29)
('NO PRIME--->', 2519)
(' f/11 PRIME NUMBER -----', 229.0)
(' N PRIME NUMBER -----', 29)
(' N= ---->', 31)
(' PRIME NUMBER -----', 2879)
(' N PRIME NUMBER -----', 31)
(' N= ---->', 33)
('NO PRIME--->', 3263)
('N NO PRIME--->', 33)
(' N= ---->', 35)
(' PRIME NUMBER -----', 3671)
('N NO PRIME--->', 35)
(' N= ---->', 37)
('NO PRIME--->', 4103)
(' f/11 PRIME NUMBER -----', 373.0)
(' N PRIME NUMBER -----', 37)
(' N= ---->', 39)
('NO PRIME--->', 4559)
('N NO PRIME--->', 39)
(' N= ---->', 41)
(' PRIME NUMBER -----', 5039)
(' N PRIME NUMBER -----', 41)
(' N= ---->', 43)
('NO PRIME--->', 5543)
(' N PRIME NUMBER -----', 43)
(' N= ---->', 45)
('NO PRIME--->', 6071)
('N NO PRIME--->', 45)
(' N= ---->', 47)
('NO PRIME--->', 6623)
(' N PRIME NUMBER -----', 47)
(' N= ---->', 49)
('NO PRIME--->', 7199)
('N NO PRIME--->', 49)
(' N= ---->', 51)
('NO PRIME--->', 7799)
(' f/11 PRIME NUMBER -----', 709.0)
('N NO PRIME--->', 51)
(' N= ---->', 53)
(' PRIME NUMBER -----', 8423)
(' N PRIME NUMBER -----', 53)
(' N= ---->', 55)
('NO PRIME--->', 9071)
('N NO PRIME--->', 55)
(' N= ---->', 57)
(' PRIME NUMBER -----', 9743)
('N NO PRIME--->', 57)
(' N= ---->', 59)
('NO PRIME--->', 10439)
(' f/11 PRIME NUMBER -----', 949.0)
(' N PRIME NUMBER -----', 59)
(' N= ---->', 61)
(' PRIME NUMBER -----', 11159)
(' N PRIME NUMBER -----', 61)
(' N= ---->', 63)
(' PRIME NUMBER -----', 11903)
('N NO PRIME--->', 63)
(' N= ---->', 65)
(' PRIME NUMBER -----', 12671)
('N NO PRIME--->', 65)
(' N= ---->', 67)
(' PRIME NUMBER -----', 13463)
(' N PRIME NUMBER -----', 67)
(' N= ---->', 69)
('NO PRIME--->', 14279)
('N NO PRIME--->', 69)
(' N= ---->', 71)
('NO PRIME--->', 15119)
(' N PRIME NUMBER -----', 71)
(' N= ---->', 73)
('NO PRIME--->', 15983)
(' f/11 PRIME NUMBER -----', 1453.0)
(' N PRIME NUMBER -----', 73)
(' N= ---->', 75)
(' PRIME NUMBER -----', 16871)
('N NO PRIME--->', 75)
(' N= ---->', 77)
(' PRIME NUMBER -----', 17783)
('N NO PRIME--->', 77)
(' N= ---->', 79)
(' PRIME NUMBER -----', 18719)
(' N PRIME NUMBER -----', 79)
(' N= ---->', 81)
('NO PRIME--->', 19679)
(' f/11 PRIME NUMBER -----', 1789.0)
('N NO PRIME--->', 81)
(' N= ---->', 83)
(' PRIME NUMBER -----', 20663)
(' N PRIME NUMBER -----', 83)
(' N= ---->', 85)
('NO PRIME--->', 21671)
('N NO PRIME--->', 85)
(' N= ---->', 87)
('NO PRIME--->', 22703)
('N NO PRIME--->', 87)
(' N= ---->', 89)
('NO PRIME--->', 23759)
(' N PRIME NUMBER -----', 89)
(' N= ---->', 91)
('NO PRIME--->', 24839)
('N NO PRIME--->', 91)
(' N= ---->', 93)
(' PRIME NUMBER -----', 25943)
('N NO PRIME--->', 93)
(' N= ---->', 95)
('NO PRIME--->', 27071)
(' f/11 PRIME NUMBER -----', 2461.0)
('N NO PRIME--->', 95)
(' N= ---->', 97)
('NO PRIME--->', 28223)
(' N PRIME NUMBER -----', 97)
(' N= ---->', 99)
(' PRIME NUMBER -----', 29399)
('N NO PRIME--->', 99)


.

Last fiddled with by Godzilla on 2019-05-10 at 19:54
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Old 2019-05-11, 06:00   #10
CRGreathouse
 
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Quote:
Originally Posted by Godzilla View Post
I discovered that with this function if I divide by the number 11 always and only every 8 or 14 sequences always generates a prime number.
As kar_bon's detective work pointed out, your function is 3n^2 - 4. The Brun sieve proves that your function cannot have this property; in particular, the fraction of n-values that can lead to primes for (3n^2 - 4)/11 is 0, and your claim would require this to be at least 1/14.

So all that remains is finding the first example where it fails.

Quote:
Originally Posted by Godzilla View Post
Code:
(' N= ---->', 59)
('NO PRIME--->', 10439)
(' f/11 PRIME NUMBER -----', 949.0)
949 = 13 * 73 is composite, so it looks like your program's jumping the gun.
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Old 2019-05-11, 07:51   #11
kar_bon
 
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Again using pfgw with the input given you can find in the output the numbers with index $a divisible by 11 as the sequence (which is A281445):
4, 7, 15, 18, 26, 29, 37, 40

So looking at the first two values (4 & 7) the next values always add 11 to this.
So you could create this pfgw input:
Code:
ABC2 3*$a^2-4 | 3*($a+3)^2-4
a: from 4 to 1000 step 11
In other words
Your numbers 3*n^2+4 are divisible by 11 for n == 4 or 7 (mod 11).

This will lead to
Code:
ABC2 3*(11*$a+4)^2-4 | 3*(11*$a+7)^2-4
a: from 0 to 50
getting this result (deleted all values divisible by 2):
Code:
3*(11*0+7)^2-4 trivially factors as: 11*13
3*(11*1+4)^2-4  trivially factors as: 11*61
3*(11*2+7)^2-4 trivially factors as: 11*229
3*(11*3+4)^2-4  trivially factors as: 11*373
3*(11*4+7)^2-4 trivially factors as: 11*709
3*(11*5+4)^2-4  trivially factors as: 11*13*73
3*(11*6+7)^2-4 trivially factors as: 11*1453
3*(11*7+4)^2-4  trivially factors as: 11*1789
3*(11*8+7)^2-4 trivially factors as: 11*23*107
3*(11*9+4)^2-4  trivially factors as: 11^2*263
3*(11*10+7)^2-4 trivially factors as: 11*3733
3*(11*11+4)^2-4  trivially factors as: 11*4261
3*(11*12+7)^2-4 trivially factors as: 11^2*479
3*(11*13+4)^2-4  trivially factors as: 11*71*83
3*(11*14+7)^2-4 trivially factors as: 11*7069
3*(11*15+4)^2-4  trivially factors as: 11*7789
3*(11*16+7)^2-4 trivially factors as: 11*9133
3*(11*17+4)^2-4  trivially factors as: 11*9949
3*(11*18+7)^2-4 trivially factors as: 11*73*157
3*(11*19+4)^2-4  trivially factors as: 11*12373
3*(11*20+7)^2-4 trivially factors as: 11*13*23*47
As mentioned the n-value 59=11*5+4 in the seq. leads not to a number only divisible by 11.
Learn to use programs used and verified over years or how to program your own correctly.
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