goldbach’s conjecture
TRANSCRIPT
Goldbach’s Number
• A Goldbach number is a positive integer that can be expressed as the sum of two odd primes.
• The expression of a given even number as a sum of two primes is called a Goldbach partition of that number. The following are examples of Goldbach partitions for some even numbers:
6 = 3 + 38 = 3 + 510 = 3 + 7 = 5 + 512 = 7 + 5..100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53
Image of first Goldbach’s Partition
The even integers from 4 to 28 as sums of two primes: Even integers correspond to horizontal lines. For each prime, there are two oblique lines, one red and one blue.
The sums of two primes are the intersections of one red and one blue line, marked by a circle. Thus the circles on a given horizontal line give all partitions of the corresponding even integer into the sum of two primes.
Goldbach’s Conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states:
• Every even integer greater than 2 can be expressed as the sum of two primes.
• Has been verified for all even numbers to 400 trillion, but not yet proved.
Some examples are:
4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 3+7,…, 100 = 53 + 47
Can every whole number grater than 2 be written as a sum of two primes?
A prime is a whole number which is only divisible by 1 and itself. Let's try with a few examples:
4 = 2 + 2 and 2 is a prime, so the answer to the question is "yes" for the number 4.6 = 3 + 3 and 3 is prime, so it's "yes" for 6 also.8 = 3 + 5, 5 is a prime too, so it's another "yes".
Second part of the definition of Goldbach’s Conjecture.
• It says that every odd whole number greater than 5 can be written as the sum of three primes.
Some examples are: 7 = 3 + 2 + 211 = 3 + 3 + 513 = 3 + 5 + 5 17 = 5 + 5 + 7
Sage Code for Goldbach’s Conjecture
We can use a simple loop to check this conjecture for the first few even numbers. For example:
for x in range(2,11): target = 2 * x for y in range(2,x+1): if (is_prime(y) and is_prime(target - y)): print(target, y, target-y)
Answer: (4, 2, 2) (6, 3, 3)(8, 3, 5)
(10, 3, 7) (10, 5, 5) (12, 5, 7)
(14, 3, 11) (14, 7, 7) (16, 3, 13) (16, 5, 11) (18, 5, 13) (18, 7, 11)
(20, 3, 17) (20, 7, 13)
References • [1] WIKIPEDIA, Goldbach’s Conjecture.
Wikipedia, https://en.wikipedia.org/wiki/Goldbach%27s_conjecture, accessed April 2016, n.p.
• [2] Wright, Chris. Number Theory, Goldbach’s Conjecture. Web. http://76.79.215.20:8000/home/pub/73/. Accessed April 2016.
• [3] Mathematical Mysteries. Goldbach’s Conjecture. Web. https://plus.maths.org/content/mathematical-mysteries-goldbach-conjecture. Accessed April 2016.
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