Can 9999 be written as the sum of two squares? or of three squares? or of four squares?
It follows from our discussions that 9999 cannot be written as the sum of two squares because , or as the sum of three squares because . But, as proved by Lagrange, every positive integer can be written as the sum of four squares: for example, 9999 can be written as or as .
Chapter 5 also contained a discussion of which right-angled triangles have sides whose lengths are all integers.
Which right-angled triangles have integer-length sides?
For a right-angled triangle to have integer-length sides, these lengths must be of the form , where k is a constant, x and y are coprime integers with one odd and the other even, and . Chapter 5 includes a table of all triples with (the so-called ‘primitive triples’) and no numbers exceeding 100.
In Chapters 2 and 5 our discussions then extended to cubes.
Must all cubes be of the form 9n, , or , where n is an integer?
The answer to this question is ‘yes’, as we proved in Chapter 2, at the end of the section on Squares. We can also express this result by saying that every cube is congruent to 0, 1, or 8 (mod 9).
Are there any integers a, b, c for which ?
By Fermat’s last theorem, discussed in Chapter 5, the equation has no non-zero solutions when . So this equation can have solutions only when at least one of the integers a, b, and c is 0: for example, and .
Can every number be written as the sum of six cubes?
In our discussion of Waring’s problem at the end of Chapter 5, we saw that 23, for example, requires at least nine cubes. However, as we remarked, every number from some point onwards can be written as the sum of seven cubes. It is not yet known whether ‘seven’ can be reduced to ‘six’.
Perfect numbers
In Chapter 3 we discussed perfect numbers, where a number n is perfect if the sum of all its proper factors (those different from n) is equal to n. The first four perfect numbers, known since the time of the Greeks, are 6, 28, 496, and 8128. The next two questions were discussed in Chapter 3.
What is the next perfect number after 8128?
After 8128 there is a large gap, and the next perfect number does not occur until 33,550,336.
Is there a formula for producing perfect numbers?
As we showed in Chapter 3, every number of the form , where is prime, is a perfect number, and all even perfect numbers can be written in this form—for example,
It is not yet known whether there are any odd perfect numbers.
Prime numbers
Prime numbers were discussed in Chapters 3 and 7. The next two questions were discussed at the beginning of Chapter 7, in the section on Two famous conjectures.
Does the list of twin primes go on for ever?
Twin primes are pairs of primes that differ by 2, and many examples are known. The twin prime conjecture is that there are infinitely many pairs of twin primes. It is generally believed to be true, but this has never been proved. Some related results are presented in Chapter 7.
Can every even number be written as the sum of two primes?
Another famous unanswered question is Goldbach’s conjecture, which asks whether every even number greater than 2 can be written as the sum of two primes. It is known to be true for all even numbers up to 400 trillion, but has not yet been proved in general.
In Chapter 7 we also saw how to construct strings of consecutive composite numbers of any desired length.
Is there a string of 1000 consecutive composite numbers?
As we saw in the section on The distribution of primes, an example of such a string of composite numbers is
In Chapter 5 we reduced the problem of deciding which numbers can be written as the sum of two squares to the corresponding problem for primes.
Which prime numbers can be written as the sum of two squares?
As we saw in the section on Sums of squares, every prime number of the form can be written in exactly one way as the sum of two squares, as can the prime 2 . However, no numbers of the form (and primes of this form, in particular) can be written as the sum of two squares.
The next two questions were discussed in Chapter 3 in connection with Mersenne and Fermat primes.
Is the number always prime when n is prime, and always composite when n is composite?
In our discussion of Mersenne primes, we saw that must be composite when n is composite. However, need not be prime when n is prime: for example, .
Are all numbers of the form , where n is a power of 2, prime?
In our discussion of Fermat primes, we saw that the first five ‘Fermat numbers’, , , , , and , are all prime. No other examples have ever been found.
The last two questions were discussed in Chapter 7, in the section on Primes in arithmetic progressions.
Are there infinitely many primes of the form ? or of the form ?
The answer to these questions is ‘yes’, and both can
