Show that every even number between 4 and 100 is the sum of two primes.

Name: Problem 13, Chapter 1: Applied Algebra: Codes, Ciphers and Discrete Algorithms
Uploaded: 2020-07-08T04:17:54-08:00
Duration: 5 min 6 s
Channel: Nick Johnson
Description: Show that every even number between 4 and 100 is the sum of two primes.

Show that every even number between 4 and 100 is the sum of...

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Key Concepts

Even Numbers

Even numbers are integers that are divisible by 2 without any remainder. They have a simple definition in number theory and play a crucial role in many conjectures and proofs, particularly when exploring patterns and the additive properties of numbers.

Prime Numbers

Prime numbers are integers greater than 1 that have no positive divisors other than 1 and themselves. They are the basic building blocks of number theory and are essential in various proofs and problems, especially in the study of sums of primes and factorization.

Goldbach's Conjecture

Goldbach's Conjecture is a famous unsolved problem in number theory proposing that every even integer greater than 2 can be expressed as the sum of two prime numbers. This concept links even numbers and primes and has been extensively tested within specific ranges, serving as a motivating example for exploring additive problems in mathematics.

Transcript

00:01 So a very famous conjecture is gold box conjecture, which states that any even integer, greater than two, can be expressed as the sum of two prime numbers.

00:16 Not necessarily distinct prime numbers.

00:19 So if you have, let's say, four, you go to two plus two.

00:23 Six is three plus three.

00:25 Eight is three plus five.

00:26 Okay.

00:27 You can always find two prime numbers added together such that they will equal any even integer greater than two.

00:36 Okay.

00:37 Well, we can test this.

00:39 So for a, let's test this conjecture on a few even numbers.

00:45 How about 10? okay? oh, 10.

00:50 So 10, there are a few.

00:51 Now, it doesn't say it's only one way to do it.

00:53 10, i mean, maybe the most obvious or maybe not the most obvious, it would be 5 plus 5, right? or we can say, well, if we want two distinct here, we can do seven plus three.

01:03 Okay.

01:04 It's 10, 5 plus 5.

01:06 So 10 definitely is the sum of two prime numbers.

01:10 How about for b or what is it still? actually, it's still part a, right? so we're still testing a few numbers.

01:18 So we have 10, and then we have 12.

01:21 Okay, so 12 should be the sum of two prime numbers.

01:27 Okay.

01:28 Now, six plus six not going to work, right? because any two prime numbers...