1: Goldbach's Conjecture is a famous conjecture that has never been proven true or false. The conjecture states that every even number, except 2, can be written as the sum of two prime numbers. For example, 16 can be written as 5+11.
1 a: Write the first six even numbers greater than 2 as the sum of two prime numbers.
1 b: Write 100 as the sum of two primes.
1 c: The number 2 is a prime number. Can an even number greater than 4 be written as the sum of two prime numbers if you use 2 as one of the primes? Explain why or why not.
I just feel like this isn't what i´m looking for and its not giving me my answer im looking for either
1a: the numbers are 4, 6, 8, 10, 12, and 14. 4 = 2 + 2; 6 = 3 + 3; 8 = 5 + 3; 10 = 5 + 5; 12 = 7 + 5.
If you don't know how to do this, it's probably because you don't understand prime numbers so well. Prime numbers are numbers only divisible by 1 and itself (so you know automatically the number will be odd - not divisible by 2!). There are infinite prime numbers, but it really helps to know a good set of them. You can Google it, but here are some in order to get you started: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, etc. See a pattern?
onto...
1b: 100 = 39 + 61. How do I know this? It really takes some guesswork, trial and error, and familiarity with prime numbers. I'm sure you'll get there soon!
1c: Absolutely not! The question specifies an even number. Now think, 'how do I get an even number as a result of addition?' The only way you can do this is add two even numbers or two odd numbers (try a few examples...). All prime numbers except 2 are odd (bc even numbers are always multiples of 2), and an even number + an odd number is always odd. Thus, 2 + a different prime number will always equal an odd number - never an even number.
Hope this helps. This is an interesting conjecture!
Thank you so much
1 a: To write the first six even numbers greater than 2 as the sum of two prime numbers, you need to check if each even number can be expressed as the sum of two primes.
Let's go through the even numbers one by one and find two primes that add up to each number:
- For 4, we can express it as the sum of 2 and 2.
- For 6, we can express it as the sum of 3 and 3.
- For 8, we can express it as the sum of 3 and 5.
- For 10, we can express it as the sum of 3 and 7.
- For 12, we can express it as the sum of 5 and 7.
- For 14, we can express it as the sum of 3 and 11.
So, the first six even numbers greater than 2 can be written as follows:
4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7
12 = 5 + 7
14 = 3 + 11
1 b: To write 100 as the sum of two primes, we need to find two prime numbers that add up to 100.
We can use a prime number calculator or a list of prime numbers to find the sum of two primes that equals 100. Checking the prime numbers below 100, we can find that 47 and 53 are prime numbers whose sum is 100.
Therefore, 100 can be written as:
100 = 47 + 53
1 c: The number 2 is a prime number, and we are asked if an even number greater than 4 can be written as the sum of two prime numbers if we use 2 as one of the primes.
No, an even number greater than 4 cannot be written as the sum of two prime numbers if we use 2 as one of the primes.
This is because if we use 2 as one of the primes, the other prime number needs to be odd, and an odd number plus an even number always results in an odd number. However, we know that every even number greater than 2 is divisible by 2 and therefore always an even number. Hence, it is not possible to sum an even number greater than 4 with 2 to get another even number.