Some Famous Problems of the Theory of Numbers And in Particular Waring's Problem; An Inaugural Lecture Delivered Before the University of Oxford
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If s,'the number of squares, is even and less than 10, the number of representations may be expressed in a very simple form by means of the divisors of n. Thus the number of representations by 4 squares, when n is odd, is 8 times the sum of the divisors of n; when n is even, it is 24 times the sum of the odd divisors; and there are similar results for 2 squares, or 6, or 8.
When s is 3, 5, or 7, the number of representations can also be found in a simple form, though one of a very difierent cha...racter. Suppose, for example, that s is 3. The problem is in this case essentially the same as that of determining the number of classes of binary arithmetical forms of determinant —n; and the solution depends on certain finite sums of the form extended over quadratic residues ;3 or non-residues y of n.
When s, whether even or odd, is greater than 8, the solution is decidedly more difficult, and it is only very recently that a uniform method of solution, for which I must refer you to some recent memoirs of Mr.
What to read after Some Famous Problems of the Theory of Numbers And in Particular Waring's Problem; An Inaugural Lecture Delivered Before the University of Oxford?
You can find similar books in the "Read Also" column, or choose other free books by G H Godfrey Harold Hardy to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
If s,'the number of squares, is even and less than 10, the number of representations may be expressed in a very simple form by means of the divisors of n. Thus the number of representations by 4 squares, when n is odd, is 8 times the sum of the divisors of n; when n is even, it is 24 times the sum of the odd divisors; and there are similar results for 2 squares, or 6, or 8.
When s is 3, 5, or 7, the number of representations can also be found in a simple form, though one of a very difierent cha...racter. Suppose, for example, that s is 3. The problem is in this case essentially the same as that of determining the number of classes of binary arithmetical forms of determinant —n; and the solution depends on certain finite sums of the form extended over quadratic residues ;3 or non-residues y of n.
When s, whether even or odd, is greater than 8, the solution is decidedly more difficult, and it is only very recently that a uniform method of solution, for which I must refer you to some recent memoirs of Mr.
What to read after Some Famous Problems of the Theory of Numbers And in Particular Waring's Problem; An Inaugural Lecture Delivered Before the University of Oxford?
You can find similar books in the "Read Also" column, or choose other free books by G H Godfrey Harold Hardy to read onlineMoreLess
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