Do there exist three integers in Arithmetic Progression whose product is prime ? If Yes, then what are the three integers and if No, then why ?

[Note: The numbers: x1, x2, x3, x4, x5, x6,........ are said to be in Arithmetic Progression if (x2 - x1) = (x3 - x2) = (x4 - x3) = (x5 - x4) = ........ and so on].

The solution is (-3, -1,  1).  The are three integers in Arithmetic Progression whose product is a "prime number": 3. 

The product of the set of three integers of Arithemtic Progression (-1, 1, 3), being -3, is also prime, but not a "prime number".  As it is prime, it may also be considered a valid solution to the problem.
 
A "prime number", by definition, is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself.  In the context of ring theory of abstract algebra, additive inverses of prime numbers are also prime.  However they are not considered "prime numbers".  Instead they are members, with "prime numbers", of a larger set called "prime elements".

Posted by Dej Mar on 2006-07-24 15:02:22