I am trying to figure out a way to measure if numbers of a given form are prime more often than expected. This would allow some way to quantify how useful certain forms are at producing large primes. I am interested in the answer to this question generally however specifically, I am looking at generalizing Mersenne Numbers with the following particular forms where the $p_i$ are distinct odd primes:
Order 1: $$2^{k_1}p_2^{k_2}-1$$ Order 2: $$p_1^{k_1}p_2^{k_2}-(p_1+p_2)$$ Order 3: $$2^{k_1}p_2^{k_2}p_3^{k_3}p_4^{k_4}-(p_3p_4+p_2p_4+p_2p_3)$$ Order 4: $$p_1^{k_1}p_2^{k_2}p_3^{k_3}p_4^{k_4}-(p_2p_3p_4+p_1p_3p_4+p_1p_2p_4+p_1p_2p_3)$$ etc...
It is clear that numbers produced by this method will not be divisible by the $p_i$, but it seems to produce large primes more often than what one might intuitively "expect". For example when $p_1=3$, $p_2=5$, and $1\leq k_i\leq10$, 40% of the 100 numbers produced are prime. If we use $1/ln(x)$ as a baseline probability we would only expect about 8.4% of the 100 numbers produced to be prime. Even if we inflate $1/ln(x)$ by $5/15$ (the ratio of numbers we expect not to be divisible by 2,3 or 5) we get 25.3%.
Is there a reason why numbers of this form seem to be prime more often than expected and is this an appropriate way to measure how well a certain form and/or sets of $p_i$ are at producing primes? What might be a better way that takes into account the numbers produced are not completely random?
? a=3;b=5;for(i=1,10, for(j=1,10, y=a+b; x=a^i*b^j-y; if(isprime(x),print(a,"^",i,"*",b,"^",j,"-",y,"=",x)) ))
3^1*5^1-8=7
3^1*5^2-8=67
3^1*5^3-8=367
3^1*5^4-8=1867
3^1*5^6-8=46867
3^1*5^8-8=1171867
3^1*5^9-8=5859367
3^2*5^1-8=37
3^2*5^3-8=1117
3^2*5^6-8=140617
3^2*5^7-8=703117
3^2*5^10-8=87890617
3^3*5^1-8=127
3^3*5^8-8=10546867
3^4*5^1-8=397
3^4*5^2-8=2017
3^4*5^6-8=1265617
3^5*5^2-8=6067
3^5*5^3-8=30367
3^5*5^8-8=94921867
3^5*5^10-8=2373046867
3^6*5^1-8=3637
3^6*5^2-8=18217
3^6*5^5-8=2278117
3^6*5^10-8=7119140617
3^7*5^2-8=54667
3^7*5^3-8=273367
3^7*5^9-8=4271484367
3^8*5^1-8=32797
3^8*5^3-8=820117
3^8*5^5-8=20503117
3^8*5^7-8=512578117
3^9*5^1-8=98407
3^9*5^2-8=492067
3^9*5^5-8=61509367
3^10*5^1-8=295237
3^10*5^2-8=1476217
3^10*5^3-8=7381117
3^10*5^8-8=23066015617
3^10*5^9-8=115330078117
The largest prime I have found of this form without trying too hard is 500 digits:
3 ^ 416 * 5 ^ 430 - 8 =
10953270873453764072182016869048685865577572728474313282647298539406621226912167384457445350880070306851250032787990008339739616770392432157060556127211791099388499068153551674197920688130962853911761257267410993254028133918782946515160568303628832641933293339172177016328514094886268133058911807159869882299664338308814657437703714727309661480035874442493113008884698464990244699917853921340830065109775134806963517063427392973118461162609561117570360482009229896593893727185786701738834381103515617