Mathematical Puzzles
Absolute maximum
The absolute value of a real number is defined as its numerical value without regard for sign. So, for example, abs(2) = abs(-2) = 2. The maximum of two real numbers is defined as the numerically bigger of the two. For example, max(2, -3) = max(2, 2) = 2. Express: (a) abs in terms of max; and (b) max in terms of abs.
Eight odd squares
Lagrange's Four-Square Theorem states that every positive integer can be written as the sum of at most four squares. For example, 6 = 2^2 + 1^2 + 1^2 is the sum of three squares. Given this theorem, prove that any positive multiple of 8 can be written as the sum of eight odd squares.
Fermat squares
By Fermat's Little Theorem, the number x = (2^(p-1) - 1)/p is always an integer if p is an odd prime. For what values of p is x a perfect square?
The absolute value of a real number is defined as its numerical value without regard for sign. So, for example, abs(2) = abs(-2) = 2. The maximum of two real numbers is defined as the numerically bigger of the two. For example, max(2, -3) = max(2, 2) = 2. Express: (a) abs in terms of max; and (b) max in terms of abs.
Eight odd squares
Lagrange's Four-Square Theorem states that every positive integer can be written as the sum of at most four squares. For example, 6 = 2^2 + 1^2 + 1^2 is the sum of three squares. Given this theorem, prove that any positive multiple of 8 can be written as the sum of eight odd squares.
Fermat squares
By Fermat's Little Theorem, the number x = (2^(p-1) - 1)/p is always an integer if p is an odd prime. For what values of p is x a perfect square?
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