I need help finding the composite number that is equal to the product of consecutive primes plus one.

for example 2 x 3 x 5......x n + 1= some number. Find out if its composite.

Someone else gave the smallest. Here are more:


2 x 3 x 5 x 7 x 11 x 13 x 17 = 510511 = 19 x 97 x 277


2 x 3 x 5 x 7 x 11 x 13 x 17 x 19 = 9699691 = 347 x 27953


2 x 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 = 223092871 = 317 x 703763


2 x 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 = 6469693231 = 331 x 571 x 34231

Note: I forgot to type in the + 1 on the left side of each, for example it should be:
2 x 3 x 5 x 7 x 11 x 13 x 17 + 1 = 510511 = 19 x 97 x 277 etc.
--Rita the dog

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Some are, some aren't. Depends on n.

2 x 3 + 1 = 7. Prime.
2 x 3 x 5 x 7 x 11 x 13 + 1 = 30031 = 59 x 509. Not prime.

Edit: That's the lowest one starting with 2, which is what I assumed you meant. As the other answer points out, 5 x 7 + 1 is an even more simple example if you don't have to start with 2.

This thing comes up in Euclid's proof that there's no largest prime. If p_n is the n-th prime, then the expression 2 x 3 x ... x p_n + 1 is not divisible by any prime up to p_n. So it's either prime, or it's a composite whose factors are larger than p_n (such as 30031 whose factors are larger than 13). Either way no matter what p_n is, it's not the largest prime.

But you don't have to know whether the expression is or isn't prime for this proof, and as I said, it depends.

How about 5 x 7 +1 = 36 (composite) ?