Concatenate the mersenne numbers $2^2-1$ to $2^n-1$ and define $f(n)$ to be the emerging number , for example $$f(6)=\color\red {3}\color\green {7}\color\red {15}\color\green {31}\color\red {63}$$
We get a prime number for $n=2,3,6,10,23$ and no other such prime numbers with less than $10\ 000$ digits exist.
Can we expect infinite many such primes ?
What is the next number $n$ such that $f(n)$ is prime ?
The following table shows the numbers $n$ upto $n=1\ 000$ , such that $f(n)$ has no prime divisor less than or equal to $611\ 953$ , the $50\ 000$ th prime. The second column shows the number of digits of $f(n)$
? t=prod(j=1,50*10^3,prime(j));k=2;s=2^k-1;k=k+1;s=c(s,2^k-1);while(k<1000,k=k+1
;s=c(s,2^k-1);if(gcd(s,t)==1,print(k," ",length(digits(s)))))
6 8
10 21
18 60
22 87
23 94
45 334
47 363
66 698
99 1538
102 1631
118 2172
150 3484
161 4006
165 4205
183 5159
202 6272
215 7097
221 7495
238 8681
239 8753
258 10187
267 10904
273 11395
282 12152
293 13111
310 14666
317 15331
334 17008
339 17518
341 17724
370 20846
406 25074
425 27463
426 27592
435 28764
453 31182
471 33696
477 34556
497 37501
498 37651
501 38104
509 39326
510 39480
514 40099
526 41986
534 43267
543 44732
551 46054
555 46723
573 49790
575 50137
579 50834
581 51184
582 51360
591 52955
609 56219
615 57328
695 73154
717 77844
723 79149
729 80464
738 82457
742 83351
743 83575
749 84926
767 89044
783 92787
803 97575
846 108276
867 113703
875 115806
891 120070
911 125508
917 127163
934 131910
935 132192
939 133323
946 135313
959 139049
966 141081
?