Find all triples $(a,b,c)$ satisfying the following conditions: (i) $a,b,c$ are prime numbers, where $a<b<c<100$. (ii) $a+1,b+1,c+1$ form a geometric sequence.
Problem
Source: China south east mathematical Olympiad 2007 problem 7
Tags: ratio, number theory, prime numbers, number theory unsolved
Ygg
12.07.2013 18:49
There are only $25$ prime numbers less then $100$, so it is not difficult to find all solutions by simple checking.
$(2, 5, 11), (2, 11, 47), (5, 11, 23), (5, 17, 53), (7, 23, 71), (11, 23, 47)$
jred
13.07.2013 13:23
Ygg wrote: There are only $25$ prime numbers less then $100$, so it is not difficult to find all solutions by simple checking.
$(2, 5, 11), (2, 11, 47), (5, 11, 23), (5, 17, 53), (7, 23, 71), (11, 23, 47)$
i don't think it's an easy job to check all the combinations of prime numbers. however, you don't get the right answer. there are 11 triples fulfilling those conditions indeed.
Ygg
13.07.2013 17:13
jred wrote: i don't think it's an easy job to check all the combinations of prime numbers. however, you don't get the right answer. there are 11 triples fulfilling those conditions indeed. Oh, I forgot that ratio can be any rational number greater than $1$. I check only for integers. There's missing triplets:
$(8, 12, 18), (18, 24, 32), (18, 42, 98), (32, 48, 72), (72, 84, 98)$
jred
24.07.2013 09:27
Ygg wrote: jred wrote: i don't think it's an easy job to check all the combinations of prime numbers. however, you don't get the right answer. there are 11 triples fulfilling those conditions indeed. Oh, I forgot that ratio can be any rational number greater than $1$. I check only for integers. There's missing triplets:
$(8, 12, 18), (18, 24, 32), (18, 42, 98), (32, 48, 72), (72, 84, 98)$
you forgot to subtract 1 from all the triplets. however, you have found all the triplets virtually.