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What's a counterexample to the statement that all odd numbers can be expressed as the sum of two primes?
There is a simple counterexample: the number 1:
- 1 is an odd number
- the first prime is 2 (not 1, see below) which is bigger than 1 so 1 cannot possibly be the sum of two primes.
There are plenty of other counterexamples:
- The sum of two odd numbers is even;
- All prime numbers except 2 are odd;
- When adding two prime numbers together, to get an odd result one of them must be even, namely 2;
- So any odd number that is 2 more than a composite number will not be expressible as the sum of two primes.
- examples: 11, 17, 23, 27, 29, 35, 37, ...
Another counterexample is the number 3:
- 3 is an odd number
- 3 can only be made by 2 + 1
- 1 is not a prime (see below)
A Prime number is a number that has exactly 2 distinct (different) factors.
The number 1 has only 1 distinct factor (the number 1) and so is not a prime number; the first prime number is 2.
What else can I help you with?
How many of the first ten whole numbers can be expressed as the sum of the two different primes?
Five of them.
What number has the most prime factors?
Hi... Every integer can be expressed as the product of prime numbers (and these primes are it's factors). Since we can multiply any integer by 2 to create a larger integer which can also be expressed as the product of primes, and this number has more prime factors than the last, we can always get a bigger number with more prime factors. Therefore, there is no definable number with the most primes (much like there is no largest number)!
Is there a pattern in the prime and composite numbers between 1 to 100?
There are some patterns, but none that can help you determine, in all cases, whether the number is a prime or not.For example: * All primes except 2 are odd numbers. However, not all odd numbers are primes. * All primes greater than 3 are of the form 6n - 1, or 6n + 1. However, not all numbers of this form are primes.
How many of the prime numbers between 10 and 20 can be written as the sum of two primes?
All prime numbers greater than 2 are odd numbers. For an odd prime to be written as the sum of two primes, one of the primes must be 2 because two odd primes will produce an even sum. 11 cannot be written as the sum of two primes. 13 = 2 + 11. 17 cannot be written as the sum of two primes. 19 = 2 + 17.
Can every odd number greater than 3 be written as the sum of two prime numbers?
No.Consider that every prime number except 2 is an odd number.Consider also that the sum of two odd numbers is always an even number.Thus, the only case in which an odd number can be expressed as the sum of two primes is when it is 2 greater than a prime number, since it can take advantage of the only even prime number, 2.For example, 21 can be expressed as 2+19, both of which are primes. However, 27 has no such two primes, since 25 is not prime (5x5=25).
How many of the first ten whole numbers can be expressed as the sum of the two different primes?
Five of them.
What is the conjecture that states that all even numbers over 2 can be expressed as the sum of two primes?
Goldbach's conjecture
What is 3136 expressed as a product of its primes?
3136 = 26*72
Student conjecture that if x is a prime number then x plus 1 is not prime. Which of the following answer is a counterexample?
x = 2 : 2 and 3 are both primes.
What number has the most prime factors?
Hi... Every integer can be expressed as the product of prime numbers (and these primes are it's factors). Since we can multiply any integer by 2 to create a larger integer which can also be expressed as the product of primes, and this number has more prime factors than the last, we can always get a bigger number with more prime factors. Therefore, there is no definable number with the most primes (much like there is no largest number)!
Can all numbers be factored into primes?
All composite numbers can.
Which set of numbers includes prime numbers only?
The set of primes would be one. The set of Mersenne primes is another. The set of all primes below 50 is another. And so on. A set which includes all primes, and only them, is the set of numbers having exactly 2 factors.
What is the longest number string that only consists of prime numbers?
The longest string of consecutive numbers that are primes is two digits long, consisting of 2 and 3 only. There are no other consecutive numbers that are primes because no even numbers greater than 2 are primes.
Is 71 and 73 twin primes?
twin primes are 2 prime numbers with a difference of 2 the prime numbers 5 and 3 are twin primes because 5 minus 3 equals 2 so 109 and 111 are not twin primes
How does the number 30 fit the goldbach's conjecture?
Goldbach's conjecture says that every even number greater than two can be expressed as the sum of 2 primes. If 30 could not be expressed as the sum of two primes, then this would disprove the conjecture. As it is, 30 can be expressed as the sum of two primes. You can express it as 11+19. Thus, Goldbach's conjecture holds in this case.
What is a composite number expressed as a product of primes?
That's known as a prime factorization.
How many prime numbers are there under 1000?
There are 168 prime numbers under 1,000 . There are also 25 primes under 100, 62 primes under 300, and 95 primes under 500
