Because 16 and 49 are both squares, we might ask:
Must the sum of the first few odd numbers, 1, 3, 5, 7, …, always be a square?
What happens if we add two squares together? The number can be written as the sum of two squares, as can the number . But the numbers 12, 14, and 15 cannot be written in this way, and we may ask:
Which numbers can be written as the sum of two squares?
However, the numbers and can be written as the sum of three squares, and the number can be written as the sum of four squares, and we can ask such questions as:
Can 9999 be written as the sum of two squares? or of three squares? or of four squares?
Squares also arise in the geometry of right-angled triangles. By the Pythagorean theorem, the lengths a, b, c of the sides of a right-angled triangle satisfy the equation (see Figure 4): for example,
and we might ask:
Which other right-angled triangles have integer-length sides?
4. Right-angled triangles.
Let’s now turn our attention to cubes.
A cube (or perfect cube) has the form , where n is an integer: for example, 343 and −216 are cubes because and . Cubes can be positive, negative, or zero, and the first ten positive cubes are
Each of these cubes is
either a multiple of 9: for example, ,
or one more than a multiple of 9: for example, ,
or eight more than a multiple of 9: for example, ,
and we might ask:
Must all cubes be of the form , where n is an integer?
An unexpected link between squares and cubes is
and we might ask:
Must the sum of the first few cubes 13, 23, 33, … always be a square?
Above we saw that the sum of two squares can be another square: for example, . We may ask whether there’s a similar statement for cubes:
Are there any integers a, b, c for which ?
Just as we can write numbers as the sum of squares, so we can also write them as the sum of cubes—for example:
So we might ask:
Can every number be written as the sum of six cubes?
I’ll answer these questions in Chapters 2 and 5 where we’ll explore squares and cubes in greater detail.
Perfect numbers
The factors, or divisors, of a given number are the positive integers that divide exactly into it, leaving no remainder: for example, the factors of 10 are 1, 2, 5, and 10. A factor that’s not equal to the number itself is a proper factor: the proper factors of 10 are 1, 2, and 5.
In Book IX of his Elements Euclid discussed perfect numbers, which were believed to have mystic or religious significance. A perfect number is a number whose proper factors add up to the original number: for example,
6 is perfect, because its proper factors are 1, 2, and 3, which add up to 6;
28 is perfect, because its proper factors are 1, 2, 4, 7, and 14, which add up to 28.
The first four perfect numbers, already known to the Ancient Greeks, are 6, 28, 496, and 8128, and we might ask:
What is the next perfect number after 8128?
and, more generally,
Is there a formula for producing perfect numbers?
We’ll explore perfect numbers in Chapter 3.
Prime numbers
As you saw earlier, a prime number is a number that has no factors other than itself and 1: for example, 13 and 17 are prime numbers, whereas is not. The first fifteen primes are
A number that is not prime (such as 14, 15, or 16) is called composite.
The number 1 is regarded as neither prime nor composite. We’ll explain why in Chapter 3, where we explore prime numbers in greater detail.
Prime numbers lie at the heart of number theory because they’re the ‘building blocks’, or ‘atoms’, of our counting system, in the sense that every number that’s greater than 1 can be obtained by multiplying primes together: for example,
In some cases we can answer difficult questions about numbers in general by first answering them for primes and then combining the results.
From the above list we see that 2 and 3 seem to be the only prime numbers that differ by 1, but that several pairs of primes differ by 2: some examples are
Such pairs are called twin primes, and larger examples include 101 and 103, 2027 and 2029, and 9,999,971 and 9,999,973. Knowing that the list of primes is never-ending, we may likewise ask:
Does the list of twin primes go on for ever?
On the other hand, we sometimes find large gaps between successive prime numbers; for example, the prime numbers 23 and 29 are separated by the five composite numbers, 24, 25, 26, 27, and 28, and the primes 113 and 127 are separated by the thirteen consecutive composite numbers from 114 to 126. But how large can these gaps be? For example:
Is there a string of 1000 consecutive composite numbers?
Another question arises when we add prime numbers. Noticing that
we might ask:
Can every even number be written as the sum of two primes?
Several prime numbers can be written as the sum of two squares: for example,
But some other primes, such as 11, 23, and 47, cannot be written like this, and we might ask:
Which prime numbers can be written as the sum of two squares?
We may also notice that some prime numbers are one less than a power of 2: for example,
But none of the following numbers of this type is prime:
Noticing that the exponents (2, 3, 5, and 7) in the first list are all prime, whereas those in the second list (4, 6, 8, 9, and 10) are all composite, we might ask:
Is the number
