Consecutive Numbers
... 9. The sum of two consecutive even integers is 106. Find the integers. 10. The sum of two consecutive odd integers is –40. Find the integers. 11. Find three consecutive integers if twice the largest is 16 less than three times the smallest. Complete solutions follow this slide, so work these first b ...
... 9. The sum of two consecutive even integers is 106. Find the integers. 10. The sum of two consecutive odd integers is –40. Find the integers. 11. Find three consecutive integers if twice the largest is 16 less than three times the smallest. Complete solutions follow this slide, so work these first b ...
was the congruence
... Here is an interesting application of congruence arithmetic: Theorem There are infinitely many primes of the form p ≡ −1(mod 4). Proof We argue in a manner similar to Euclid’s proof that there are infinitely many primes. € Suppose there are only finitely many primes of the desired form. List them: ...
... Here is an interesting application of congruence arithmetic: Theorem There are infinitely many primes of the form p ≡ −1(mod 4). Proof We argue in a manner similar to Euclid’s proof that there are infinitely many primes. € Suppose there are only finitely many primes of the desired form. List them: ...
Full text
... integers in which any adjacent pair represents a solution. According to ([3], p. 56), every chain of solutions to our equation must contain an a-value in the set {0, ±1} or a Z)-value in the set {0, ±1}. The only solutions (a, b) having this property are ±(1, 1), ±(1, 2 ) , and ±(2, 1). So, except f ...
... integers in which any adjacent pair represents a solution. According to ([3], p. 56), every chain of solutions to our equation must contain an a-value in the set {0, ±1} or a Z)-value in the set {0, ±1}. The only solutions (a, b) having this property are ±(1, 1), ±(1, 2 ) , and ±(2, 1). So, except f ...
Prove if n 3 is even then n is even. Proof
... Incorrect reasoning occurs in the following cases when the propositions are assumed to be tautologies (since they are not). • Fallacy of affirming the conclusion • [(p q) q] p ...
... Incorrect reasoning occurs in the following cases when the propositions are assumed to be tautologies (since they are not). • Fallacy of affirming the conclusion • [(p q) q] p ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.