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How many twin primes are there?
There are an infinite number of twin primes. This is true, but as yet there has not been published a valid proof of it. Perhaps soon someone will publish a valid proof.
What is the smallest prime?
The smallest positive prime is 2, but there is no smallest negative prime.One way to show this is to demonstrate that the cardinality of the set of negative primes is countably infinite (the proof will be similar to that for positive primes. Hint: use proof by contradiction. Assume a finite set of negative primes and derive a contradiction).Let me know if you'd like me to write up a formal proof that there is no smallest negative prime. I'll be happy to dig it out of one of my old homework sets! :)
Can an odd number be written as the sum of two primes?
If someone says it can't, here's a counter-example. 2 + 3 = 5 This is not a proof.
How do you prove space as infinite mathematically?
There is no mathematical proof that space is infinite. All we know is that there is an expanding limit to what we can see.
What is euclid's theory about prime numbers?
That there are an infinite number of prime numbers. Before we look an explanation or proof, we must agree on some points 1. The term number means whole number or integer 2. A prime number is any number that has only 2 factors (1 and itself). 3. All numbers are either prime or the product of 1 or more primes; try and find a number that you cannot generate as the product of primes (e.g. 8 = 2x2x2; 36 = 2x2x3x3). Now: If you take any two or more prime numbers and find their product the resulting number will have the prime numbers used as factors. However, if you add 1 to the number then the prime numbers you used to produce this number will now no longer be factors of this new number. Example 2,3,5 (first three prime numbers) 2x3x5 = 30 30 +1 =31 - now 2,3 and 5 are not factors as you will always have a remainder of 1 if you divide by any of the three original prime factors (2,3 or 5). If you take all of the known prime numbers and find the product of all of these prime numbers we get a new number (call it Product of Primes or PP), PP will have all the know primes as its factors. If we now add one to PP (PP + 1=N) we will get a number, N, that will have none of the known primes as a factor. If we say that the highest value prime number known (that we used to generate PP) is Pi then N must either be prime or have a prime factor greater than Pi and thus Pi is not the highest prime number. Therefore there are an infinite number of prime numbers.
How many twin primes are there?
There are an infinite number of twin primes. This is true, but as yet there has not been published a valid proof of it. Perhaps soon someone will publish a valid proof.
What is the mathematical proof to show that the number of prime numbers is infinite?
The proof is by contradiction: assume there is a finite number of prime numbers and get a contradiction by requiring a prime that is not one of the finite number of primes. Suppose there are only a finite number of prime numbers. Then there are n of them.; and they can all be listed as: p1, p2, ..., pn in order with there being no possible primes between p(r) and p(r+1) for all 0 < r < n. Consider the number m = p1 × p2 × ... × pn + 1 It is not divisible by any prime p1, p2, ..., pn as there is a remainder of 1. Thus either m is a prime number itself or there is some other prime p (greater than pn) which divides into m. Thus there is a prime which is not in the list p1, p2, ..., pn. But the list p1, p2, ..., pn is supposed to contain all the prime numbers. Thus the assumption that there is a finite number of primes is false; ie there are an infinite number of primes. QED.
What is the smallest prime?
The smallest positive prime is 2, but there is no smallest negative prime.One way to show this is to demonstrate that the cardinality of the set of negative primes is countably infinite (the proof will be similar to that for positive primes. Hint: use proof by contradiction. Assume a finite set of negative primes and derive a contradiction).Let me know if you'd like me to write up a formal proof that there is no smallest negative prime. I'll be happy to dig it out of one of my old homework sets! :)
Can an odd number be written as the sum of two primes?
If someone says it can't, here's a counter-example. 2 + 3 = 5 This is not a proof.
Why have there been no direct proofs of the intinity of primes and naturals after 2000 years of study?
Sorry, but both of these have direct and relatively simple proofs. To prove the infinity of primes, let p1, p2, p3, ... pn be the first n primes. Form the product p1 x p2 x p3 ... x pn. Clearly it is divisible by all the prime numbers up to pn. Then add 1 to that sum. The new value cannot be divisible by any of the preceding primes so it must be a new prime number. Because you can do this for any value of n, the number of primes must be infinite. The proof of the irrationality of the square root of 2 is similarly easy and should be available in any high-school math book at the level of Algebra 2 or above.
How many prime numbers exist?
There are an infinite amount of prime numbers. The first to have proven this was Euclid. Here are the general lines of his proof: # Suppose there are n prime numbers overall. # Let N be a common multiple of all these primes. # Is N+1 prime? If it is, we have found a new prime. # Suppose N+1 is not prime. Thus, there exists a prime number p which divides N+1 evenly. If p is one of the primes dividing N, it also divides 1 evenly, which is impossible. Thus, p is not one of the n primes, and we found a new prime.
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.
What is the proof about multiplying twin primes and adding 1?
The question depends on what it is that you want to prove!
What 2 prime numbers make 123?
There is no pair of prime numbers that sum to 123.Proof :Any two odd primes will sum to an even number, thus excluding 123 which is odd.Since 2 is the only even prime, the only possible solution is 2 + 121 = 123.But, 121 is not prime because it is 11 x 11.A similar proof shows that there is no pair of primes whose difference is 123.There are two primes that multiply to 123 : - 3 and 41.
How do you prove space as infinite mathematically?
There is no mathematical proof that space is infinite. All we know is that there is an expanding limit to what we can see.
What is euclid's theory about prime numbers?
That there are an infinite number of prime numbers. Before we look an explanation or proof, we must agree on some points 1. The term number means whole number or integer 2. A prime number is any number that has only 2 factors (1 and itself). 3. All numbers are either prime or the product of 1 or more primes; try and find a number that you cannot generate as the product of primes (e.g. 8 = 2x2x2; 36 = 2x2x3x3). Now: If you take any two or more prime numbers and find their product the resulting number will have the prime numbers used as factors. However, if you add 1 to the number then the prime numbers you used to produce this number will now no longer be factors of this new number. Example 2,3,5 (first three prime numbers) 2x3x5 = 30 30 +1 =31 - now 2,3 and 5 are not factors as you will always have a remainder of 1 if you divide by any of the three original prime factors (2,3 or 5). If you take all of the known prime numbers and find the product of all of these prime numbers we get a new number (call it Product of Primes or PP), PP will have all the know primes as its factors. If we now add one to PP (PP + 1=N) we will get a number, N, that will have none of the known primes as a factor. If we say that the highest value prime number known (that we used to generate PP) is Pi then N must either be prime or have a prime factor greater than Pi and thus Pi is not the highest prime number. Therefore there are an infinite number of prime numbers.
When is proof number used?
proof number
