Term
| Prove the square root of 2 is an irrational number. |
|
Definition
|
|
Term
| Prove if a divides b and b divides c then a divides c. |
|
Definition
|
|
Term
| Prove Every odd integer is the difference of two perfect squares. |
|
Definition
|
|
Term
| Prove The number 100...01 (with 3n-1 zeros where n is an integer larger then 0) is composite. |
|
Definition
|
|
Term
| Prove If r1 and r2 are distinct roots of the polynomial p(x) = x2 + b x + c, then r1 + r2 = - b and r1 r2 = c. |
|
Definition
|
|
Term
| Prove If a divides b and a divides c then a divides b + c. (Here a, b, and c are positive natural numbers and the definition of divisibility is given above.) |
|
Definition
|
|
Term
| Prove If a is an integer, divisible by 4, then a is the difference of two perfect squares. |
|
Definition
|
|
Term
| Prove If a and b are real numbers, then a2 + b2 >= 2 a b. |
|
Definition
|
|
Term
| Prove The sum of two rational numbers is a rational number. |
|
Definition
|
|
Term
| Prove If r1, r2, r3 are three distinct (no two the same) roots of the polynomial p(x) = x3 + b x2 + c x+ d, then r1 r2 + r1 r3 + r2 r3 = c. |
|
Definition
|
|
Term
| Prove There are no positive integer solutions to the diophantine equation x2 - y2 = 1. |
|
Definition
|
|
Term
| Prove There are no rational number solutions to the equation x3 + x + 1 = 0. |
|
Definition
|
|
Term
| Prove There are no rational number solutions to the equation x3 + x + 1 = 0. |
|
Definition
|
|
Term
| There are no positive integer solutions to the diophantine equation x2 - y2 = 10. |
|
Definition
|
|
Term
| There is no rational number solution to the equation x5 + x4 + x3+x2+ 1 = 0. |
|
Definition
|
|
Term
| If a is a rational number and b is an irrational number, then a+b is an irrational number. |
|
Definition
|
|
Term
| If x and y are two integers for which x+y is even, then x and y have the same parity. (Two integers are said to have the same parity if they are both odd or both even.) |
|
Definition
|
|
Term
| If x and y are two integers whose product is even, then at least one of the two must be even. |
|
Definition
|
|
Term
| If x and y are two integers whose product is odd, then both must be odd. |
|
Definition
|
|
Term
| If a and b a real numbers such that the product a b is an irrational number, then either a or b must be an irration number. |
|
Definition
|
|
Term
| If a is an integer, then a is not evenly divisible by 3 if, and only if, a2 -1 is evenly divisble by 3. |
|
Definition
|
|
Term
| A postive integer n is evenly divisible by 3 if, and only if, the sum of the digits of n is divisble by 3. |
|
Definition
|
|
Term
| If a is an integer, then a is not evenly divisible by 5 if, and only if, a4 -1 is evenly divisble by 5. |
|
Definition
|
|
Term
| For two integers a and b, a+b is odd if, and only if, exactly one of the integers, a or b, is odd. |
|
Definition
|
|
Term
| For two integers a and b, the product ab is even if and only if at least one of the integers, a or b, is even. |
|
Definition
|
|
Term
| A postive integer n is evenly divisible by 9 if, and only if, the sum of the digits of n is divisble by 9. |
|
Definition
|
|
Term
| A postive integer n is evenly divisible by 11 if, and only if, the difference of the sums of the digits in the even and odd positions in n is divisible by 11. |
|
Definition
|
|
