I came across this conjecture:
Up to which $p$ has this conjecture be verified ?
Are there intermediate results related to this conjecture ?
The conjecture can be formulated in this way :
For every prime $p>2$, there is a prime $q$, such that $(p-1)q-1$ is prime.
The following PARI/GP-program shows, that for every $p$ with $2<p<10^8$, there is a prime $q\le 1187$ doing the job.
? maxi=0;p=2;while(p<10^8,p=nextprime(p+1);q=1;gef=0;while(gef==0,q=nextprime(q+
1);u=(p-1)*q-1;if(isprime(u)==1,gef=1;if(q>maxi,maxi=q;print(p," ",q," ",u)))
))
3 2 3
19 3 53
73 5 359
109 11 1187
139 19 2621
179 23 4093
467 29 13513
1229 71 87187
2447 83 203017
2819 269 758041
8699 281 2444137
24419 311 7593997
57977 443 25683367
266117 503 133856347
374399 641 239989117
711089 653 464340463
4099493 719 2947534747
4313873 821 3541688911
8466209 881 7458729247
13187129 1031 13595928967
52172843 1187 61929163453
?
Update : The conjectrue remains true for $2<p<10^9$ and a prime $q\le 1709$ does the job in that range.
? while(p<10^9,p=nextprime(p+1);q=1;gef=0;while(gef==0,q=nextprime(q+1);u=(p-1)*
q-1;if(isprime(u)==1,gef=1;if(q>maxi,maxi=q;print(p," ",q," ",u)))))
103784147 1193 123814486177
112564253 1361 153199946971
174454277 1367 238478995291
280721033 1511 424169479351
623020787 1613 1004932527817
855177437 1697 1451236108891
929408729 1709 1588359516151
?