![]() ![]() Modern computer calculations have shown this conjecture to be true for integers up to at least 4 × 10 18, but still no general proof has been found. Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers. All known perfect numbers are even it is unknown whether any odd perfect numbers exist. An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2.Īll prime numbers are odd, with one exception: the prime number 2. The even numbers form an ideal in the ring of integers, but the odd numbers do not-this is clear from the fact that the identity element for addition, zero, is an element of the even numbers only. Then an element of R is even or odd if and only if its numerator is so in Z. But an even plus an odd? That is odd.− 2 ⋅ 2 = − 4 0 ⋅ 2 = 0 41 ⋅ 2 = 82 may be called odd.Īs an example, let R = Z (2) be the localization of Z at the prime ideal (2). That aren't multiples of two, that aren't even. So let's just remind ourselves, even numbers, they're multiples Out many, many, many more to make sure that youįeel good about this, but an even and an odd isĪlways going to give you an odd. Have an even plus an odd, and I've only tried out two cases here, but I encourage you to try Well what about, whatĪbout four plus three? Four plus three, well Is that going to be? Well that's going to be equal to three. What about an even, what about an even plus an odd? An even plus an odd. And once again, I encourage you to try out as many numbers as you can Plus an even I get an even, but also if I have an odd plus an odd, then I also get an even number. Seems to be forming here is that if I have an even What if I add 15 plus seven? These are both odd numbers, when I add them together I get 22. What is one plus three? Well that's equal to four. Now what happens if youĪdd an odd and an odd? What happens if you add an odd number plus another odd number? Well let's try. Number that's a multiple of two, it makes sense that the sum is going to be a multiple of two. That's a multiple of two, and you add it to another Then another even number, add them together and you'll I encourage you to keep trying, pick some even number, So it looks like all theĮxamples that I've done so far, when I add an even to anĮven, I get an even number. why don't we do a simpler number, plus four, that is equal to 160. If I say 14 plus four, well that equals 18. ![]() ![]() If I say two plus six, that is going to be equal to eight. It going to be even or odd? Well you could take some examples. If I take an even number, an even number, plus another even number. So let's think, in particular, actually let me move over Happens when we operate on odd or even numbers. Odd or an even number is, let's think about what The easiest way I think about odd, it's not a multiple of two, it's not even. The doughnuts whole, one person would have to get one extra doughnut than the other person. You can't split itĮvenly between two people without breaking up a doughnut. why is it called odd? Well, it'd be hard to The number 1,441 has a one in the ones place. Look at the ones place to spot an even number, youĬan also look at the ones place to spot an odd number. So what are some examples of odd numbers? Well one, three, five, seven, nine, and of course you can So what are odd numbers? Well one way to think about them is they're the numbers that aren't even. Whole thing is going to be an even number. You have an even number, you are looking, the Ones place of it and see, we have a zero there. a number like 150 is an even number, and we can look at the And then that actually makes our looking at the ones So zero is a multiple of two, and so zero, for sure, is an even number. How is it a multiple of two? Well, zero times two is equal to zero. And the number zero is even because it is a multiple of two. That's an interesting one, why some people sometimes say, "Well is this one really even?" And that's the number zero. The number 5,977,354, 5,977,354, well that's an even number,īecause the ones place is even. Number 32 is an even number because in the ones place you have a two. An easy way to spot an even number is that it's ones place We could keep going on, and on, and on, and on. All of the multiples of two, these are even numbers. Four doughnuts, you could split thatĮvenly with two people. ![]() If you had two people, you could give oneĭoughnut to each person. So what does it mean to be even? Well an even number is one where if you had that many doughnuts, you could split it evenlyīetween two people. You to in this video is a way to classify numbersĪs either being even, or being odd. ![]()
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