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 2019-05-03, 23:06 #1 Godzilla     May 2016 16210 Posts 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)
 2019-05-05, 12:58 #2 Godzilla     May 2016 2×34 Posts 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?
 2019-05-05, 14:00 #3 kar_bon     Mar 2006 Germany 2×11×131 Posts 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*$a^2-4 a: from 1 to 10000 - run this by calling "pfgw -l test.txt" - got a result in seconds, see the created file "pfgw-prime.log" 2019-05-05, 15:27 #4 Godzilla May 2016 2428 Posts Quote:  Originally Posted by kar_bon Frst: Why such cryptic function? What is "a-1+a-1-a-1+a-1+a"? Second: - download a version of PFGW for your system - create a text-file "test.txt" with Code: ABC2 3*$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?  2019-05-05, 16:18 #5 a1call "Rashid Naimi" Oct 2015 Remote to Here/There 22×503 Posts 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  2019-05-06, 06:47 #6 kar_bon Mar 2006 Germany 2·11·131 Posts Can't help either, no MAC. But: You tried in command line "pfgw ...", why? Try "pfgw64 ..."?  2019-05-06, 07:15 #7 M344587487 "Composite as Heck" Oct 2017 14278 Posts 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  2019-05-09, 13:55 #8 Godzilla May 2016 2428 Posts 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? 2019-05-10, 19:48 #9 Godzilla May 2016 2×34 Posts Quote:  Originally Posted by Godzilla 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 2019-05-11, 06:00 #10 CRGreathouse Aug 2006 32×5×7×19 Posts Quote:  Originally Posted by Godzilla 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 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.  2019-05-11, 07:51 #11 kar_bon Mar 2006 Germany 2·11·131 Posts 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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