Discrete Mathematics Problems
... 27. Prove that any positive integer can be factored into primes. (Note it is more difficult to prove this factorization is unique, don’t worry about that for this problem). 28. [2] Prove that n3 + 2n is divisible by 3 for all positive integers n. ...
... 27. Prove that any positive integer can be factored into primes. (Note it is more difficult to prove this factorization is unique, don’t worry about that for this problem). 28. [2] Prove that n3 + 2n is divisible by 3 for all positive integers n. ...
09 Something Different
... but the plot does show there is definitely some sort of correlation between pairs of numbers ...
... but the plot does show there is definitely some sort of correlation between pairs of numbers ...
The Great Pyramid of Gizah was built for Pharaon Chufu (known
... A point marked after this rule is called by him the golden point, and the relation between the longer part and shorter one - the golden mean (or: golden ratio, mean ratio). It seems that the same problem and name were treated by earlier Greek mathematicians, especially Phytagoras (580-500 BC). The G ...
... A point marked after this rule is called by him the golden point, and the relation between the longer part and shorter one - the golden mean (or: golden ratio, mean ratio). It seems that the same problem and name were treated by earlier Greek mathematicians, especially Phytagoras (580-500 BC). The G ...
Full text
... all other cases a factor -Yz/ is missing. Therefore, the reference term q$ of (3.3) is changed as stated in (3.4). Putting these results together, we have proved Proposition 2 given in the first section, because the elements of Q(n) u q-Q are all the terms of Sn, and the multiplicity of a term with ...
... all other cases a factor -Yz/ is missing. Therefore, the reference term q$ of (3.3) is changed as stated in (3.4). Putting these results together, we have proved Proposition 2 given in the first section, because the elements of Q(n) u q-Q are all the terms of Sn, and the multiplicity of a term with ...
Prop. If n is an integer, then 3 | (n 3 − n). Proof. By the Division
... for all integers n. This would be correct, and a uniform proof (for all primes p) follows from Fermat’s Little Theorem. Remark. You may be tempted to conjecture the following. If a is not a prime number, then a - (na − n) for some integers n. You could check this conjecture for all composite numbers ...
... for all integers n. This would be correct, and a uniform proof (for all primes p) follows from Fermat’s Little Theorem. Remark. You may be tempted to conjecture the following. If a is not a prime number, then a - (na − n) for some integers n. You could check this conjecture for all composite numbers ...
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.