Jenny's phone number is 8675309, according to the song. Here are some problems about the number 8675309.
1. 8675309 is prime. How many prime phone numbers are there if phone numbers are 7 digits and cannot start with 0?
2. How many prime phone numbers are there if phone numbers are 7 digits and can start with 0?
3. 8675309 is the 582161st prime, and 582161 is prime, too. Therefore, 8675309 is a
superprime. How many superprimes are ≤8675309?
4. 8675311 is prime. Therefore, 8675309 and 8675311 are twin primes. How many twin prime pairs (n, n+2) are there such that n≤8675309?
I could have sworn that when this was in the queue a combined 3 and 4 was asked for also: a count of superprimes that are also the bottom prime of a twin. In any case, I've included that here also.
10 P=1
20 while P<9999999
30 P=nxtprm(P)
40 inc Ct
50 if P>1000000 then inc Ct7d
60 if prmdiv(Ct)=Ct and P<=8675309 then inc Supct
70 if prmdiv(P+2)=P+2 and P<=8675309 then inc Twinct
80 if prmdiv(Ct)=Ct and prmdiv(P+2)=P+2 and P<=8675309 then inc Suptwinct
150 wend
160 print Ct1,Ct7d1,Supct,Twinct,Suptwinct
finds
664579 586081 47753 52196 4281
meaning
664579 primes of up to 7 digits (puzzle part 2),
586081 primes with exactly 7 digits (part 1),
47753 superprimes in the given range,
52196 twin prime pairs,
4281 twin prime pairs where the lower number is a superprime.
BTW, adding line
90 if prmdiv(Ct)=Ct and prmdiv(P2)=P2 and P<=8675309 then inc Suptwinct2
finds that 4431 superprimes in the range are the upper number in a twin prime pair.
in fact
100 if prmdiv(Ct)=Ct and prmdiv(P2)=P2 and prmdiv(P+2)=P+2 and P<=8675309 then inc Suptwinct3
finds that in two cases a superprime was between two other primes. Actually however there's only one such case: 5: it's the 3rd prime and both 2 and 7 are also prime. The code erroneously counted 3, as the 2nd prime (ok), but it counted 1 as being prime, due to the nature of the UBASIC interpreter giveing 1 as the lowest prime factor of 1. This also affects the count of twin prime pairs where the upper number is a superprime, as it would look at 1 (as that would be p2) also and declare it prime, so that count should be 4430 rather than 4431.

And, I see from tomarken's comment that the super prime count itself, 47753 above, is also off by 1. I see now that this is the result of 2 being counted, as it is the 1st prime and the software is again counting 1 as prime.

Posted by Charlie
on 20140226 17:07:50 