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How will you prove that there will be many more prime numbers from a given set of primes?
As the question is phrased, it is not possible.
You need to start with the set of all primes up to some value. Then, you can construct a new prime by multiplying all the numbers in the set together and adding 1.
If you divide this new number by any number in the existing list, it will leave a remainder of 1. So none of them will divide it evenly. Since the number has no smaller prime factor, it is a prime.
The need to have a complete list of primes up to some value can be seen by considering the set {3, 5}. The above method would then suggest 16, which is not divisible by 3 nor by 5. But it is not a prime, either.
What else can I help you with?
Use the fact that there are infinitely many primes to prove that there are inifitely many non prime numbers?
For every prime number p greater than 2, p + 1 is composite.
Is there a infinite number of prime numbers?
Yes. To prove this, we must first assume the answer to be no. If there are a finite number of primes, there must be a largest prime. We'll call this prime number n. n! is n*(n-1)*(n-2)*...*3*2*1. n!, therefore, is divisible by all numbers smaller than or equal to n. It follows, then that n!+1 is divisible by none of them, except for 1. There are two possibilities: n!+1 is divisible by prime numbers between n and n!, or it is itself prime. Either way, we have proved that there are prime numbers greater than n, contradicting our initial assumption that primes are finite, proving that the number of primes is infinite.
Is there an infinite number of prime number?
Yes. To prove this, we must first assume the answer to be no. If there are a finite number of primes, there must be a largest prime. We'll call this prime number n. n! is n*(n-1)*(n-2)*...*3*2*1. n!, therefore, is divisible by all numbers smaller than or equal to n. It follows, then that n!+1 is divisible by none of them, except for 1. There are two possibilities: n!+1 is divisible by prime numbers between n and n!, or it is itself prime. Either way, we have proved that there are prime numbers greater than n, contradicting our initial assumption that primes are finite, proving that the number of primes is infinite.
What is the highest prime number?
There is no highest prime number. Given any set of prime numbers, you can prove that there is at least one prime number that is not in that set. Here's how. First, recall that every natural number is either a prime number, or is a composite number and thus the product of some series of prime numbers. Now, suppose you have a set of n prime numbers, P1 .. Pn. Multiply them all together and add one. Call this number Q. Q is either prime or composite. If Q is prime, then it is obviously not one of P1 .. Pn, and thus is a new prime number not in that set. If Q is composite, then there must be a list of primes that evenly divide into it. Because Q is one greater than the product of P1 .. Pn, dividing by any of those primes will have a remainder of 1. So there must be some new prime, call it R, that divides evenly into Q. In either case, Q or R is a new prime. Now suppose that you have some potential highest prime. Enumerate all of the primes lower than it, and follow the above procedure with that set of primes. You will end up with a new prime not in that list. Since you have listed all primes less than your purported highest prime, any new prime number must be greater than your highest prime. Thus, there is no highest prime.
Is the set of prime numbers is well defined or not and why?
The set is well defined. Whether or not a given integer belongs to the set of prime numbers is clearly defined even if, for extremely large numbers, it may prove impossible to determine the status of that number.
Why would you divide 61 using prime numbers only?
To prove that 61 is a prime number.
Since numbers are relatively prime if their greatest common factor is 1 how do you prove that successive odd numbers are prime?
Find their GCF.
Why is there no direct proof of the infinity of primes naturals sqrt2 after 2000 years?
A direct proof of the infinity of primes would require what is essentially a formula to calculate the Nth prime number; such a formula isn't even guaranteed to exist. It's possible to formulate a proof of the infinity of primes that would be, in a sense, direct. A direct proof that the square root of 2 is irrational is impossible, because the irrational numbers aren't defined in any direct way - just as the real numbers which aren't rational. So to prove that the square root of 2 is irrational, we have to prove that it's not rational, which requires indirect techniques.
How are prime and composite numbers used in everyday jobs?
Prime numbers and composite numbers are not used in daily jobs. However they are used by scientists to prove theorems.
Is the set of prime numbers well defined?
The set is well defined. Whether or not a given integer belongs to the set of prime numbers is clearly defined even if, for extremely large numbers, it may prove impossible to determine the status of that number.
How do you prove that 13 is a prime?
Because like all prime numbers it has only two factors which are itself and one
How do you prove that there is a prime between n and 2n?
To be pedantic, the question should say "for all n >= 2". A detailed proof is given here: http://mathforum.org/library/drmath/view/51527.html The proof is quite long, but it only uses properties of logarithms, exponents, and the binomial theorem, so if you know about these and have enough mental stamina, you can probably make sense of it.
