There is only one 1 in the entire set of **natural numbers**, which are 2 and 3. The rationale is straightforward. One of **any pair** of successive integers is always an even number, because as everyone knows, an even number other than 2 isn't prime. So, there is only one 1 in the set.

Now let's take a look at **some other integers** between 1 and 100. They are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

You should be able to see that every integer between 1 and 100 is used for exactly one pair of primes. There are no missing values: if you miss **any integers**, then there will be a pair of primes that you have not found yet, which is impossible. This means that all the integers from 1 to 100 are necessary to prove that two consecutive integers are always a pair of primes.

Furthermore, we can conclude that there are actually two different proofs that any two consecutive integers are a pair of primes. If we combine these two proofs together, then we can say that there are a total of 100 proof steps required to show this fact about prime numbers.

Easy to understand remarks The first, smallest, and only odd prime gap is the one-size-fits-all gap between 2, the first even prime number, and 3, the first odd prime number. The other prime gaps are all equal. There is just one set of successive gaps that are 2 in length: the gaps g2 and g3 between the primes 3, 5, and 7. These are the only two pairs of consecutive prime numbers of **the same type**; they are always odd or always even, but not both at the same time.

Irregular sequences are those that do not fall into any other category. Some examples of **irregular sequences** include: shortest path lengths on graphs, hole frequencies in puzzles, how many times does a person's name appear in The Lord of the Rings.

This method of proof involves assuming what you want to prove is false and then showing that this assumption leads to a conclusion that cannot be true. For example, let's say you want to prove that 4 is a prime number. You could simply assume that 4 is not a prime number and show that this assumption leads to a conclusion that can't be true - namely, that 2 is also not a prime number. Since it can be shown that no number except 1 and itself can divide into both 2 and 4, we know that at least one of **these assumptions** must be wrong! Thus, 4 is a prime number after all.

Prime numbers are whole numbers bigger than one with **only two factors**: one and the number itself. Prime numbers can only be divided by the number 1 or by themselves. For example, the first few prime numbers are 2, 3, 5, 7, and 11. These numbers cannot be divided by **any other number** except 0.

All other numbers are composite numbers - that is, they can be written as products of smaller numbers. The factors of a composite number can be positive or negative integers. Examples of composite numbers are 4.9, 2150, 2593, and 2832.

Every integer greater than or equal to 2 is either prime or composite. There are only two exceptions to **this rule**: 2 is prime but also composite (because it's divisible by itself), and 1 is prime but also odd (because it's not divisible by 2 or by 1).

In fact, all integers greater than or equal to 2 are either prime or squareful (i.e., can be divided by some number other than 1 or itself).

It's easy to check whether an integer is prime: just divide it by **all the numbers** from 2 up to but not including **its own size**. If no division yields a remainder of 0, then it's prime. Otherwise, it's not prime.

A prime number can only have one factor: itself. Because any even number contains 2 as a component, the number cannot be prime if it has itself, 2 and 1 as factors. Because 2 is an even number with **only itself** and 1 as components, it is the only even number that is a prime.

A prime number is an integer or whole number with just two components, one and itself. In other words, a prime number can only be divided equally by 1 and by itself. Prime numbers must also be bigger than one. For example, the number three is a prime number because it cannot be divided equally by any integer other than one and three. The unit step (also called jump, skip, or leap) is a special case of a divisor that does not divide **its multiplicand**; since one can always make a new multiplicand equal to the product of the old one by this value, there can only be one prime number greater than one. Therefore, two is the only prime number in its own family.

Since two is the only prime number in its own family, it follows that every other number besides two can be divided by **at least one other number** other than one. That means that every number except for two is a composite number - that is, not prime. Two is the only prime number in its own family because if there were more than one, they would have **a common factor** other than one, which is impossible.

As an example, let's say that you want to know if 9 is a prime number. It can be divided by 2, 3, and 9. Since none of these things divide it evenly, it is not prime.

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47, As a result, there are 15 prime numbers between 1 and 50. Because other integers between 1 and 50 can also be split, there are actually 16 prime numbers between 1 and 50.