1comm'on this one is really easy
Suppose we take the first n primes,
multiply them together, and add one. Does this number have to be prime? Give a proof or a
counterexample. be confident in your answer
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13 Answers
11Well, we know that a prime number can be expressed as 6k+1. So the product of first n primes (which contain 2 and 3), it will be a multiple of 6. Add one to it and you have a prime [1]
HOPE I'M NOT BLABBERING[4]
1then you'll have to prove that a prime number can be expressed as 6k+1
it may be a well known fact but i want a pure proof
and btw the actual proof is simpler
i read that post about these primes and really i din get how it can be expressed as 6K+1
11I didn't get it too. I just blabbered and bhaiyyah said it was correct [11]
1either evryone knows or they re too content with the 6k+1 thing
but hey you must have come across
Euclids proof that there are infinite no. of primes if you know that then this is too easy. in that case you can neglect this thread [1]
1correct but i didnt understand what breaks down
surely you don mean euclids proof is wrong coz it is slightly different from the prob given above the prob is often used to test the understanding of the poof itself
341what breaks down is the conjecture that p#+1 is a prime where p# is the product of all primes less than or equal to p.
Euclid's proof deals with proving that the set of primes is not finite. So, if there is a last prime number pn consider pn#+1.
If it is a prime we are done.
If its not, the fundamental theorem of algebra says that it has a prime factor. But none of the primes in the set (2,3,5,...,pn) divides pn#+1. Hence we have produced at least one prime distinct from the given set and since n is arbitrary, the number of primes is without bound.
1ya thanx for elaborating
though i have read the proof in detail earlier
1according to the fundamental theorem of arithmetic a number X can be expressed as a product of primes less or = X. let product of n numbers be N
so N+1 is not divisible be any primes <N. thus we can say it itself is a prime
62rahul you are giving the proof that shows that there are infinitely many primes.
You have not understood philip's question.