Sort a list in descending order using the Function in the complete example code. The output will show the sorted list. Another option to sort the list in descending order is to use the LINQ Library. Question: For Project Euler , the task requires me to analyze a descending order consisting of numbers with three […]

# Category: Number theory

## Sum of Powers of 2, 3, and 5 Yielding Integer Values

The task at hand is to demonstrate that any positive integer can be expressed as a combination of unique powers of two. Hence, we can observe that if we add $2^0$ to $k$, we get $k+1$. If $k+1$ happens to be an odd number, we can represent it as a sum of distinct powers of […]

## Identifying Leap Years in 2000

The number of non-leap years under the newer Gregorian calendar, which is explained in your question, needs to be determined. An alternative method to find the answer is to calculate $$ {2000over4}-{2000over100}+{2000over400}-1 $$ This approach assumes that every fourth year is a leap year and includes the year 2000. Solution 1: Before the adoption of […]

## Finding Inverses Using Exponentiation Method: A Guide

An example of this can be seen in the following equations: $begin{cases}&log_3x=yiff 3^y=x\&3^y=xiff log_3x=yend{cases}$. It is now evident that $f(x):=log_ax$ and $g(x):=a^x$ are inverse functions. This means that $fcirc g(x)=f(g(x))=log_a(a^x)=x=3^{log_ax}=g(f(x))=gcirc f(x)$. To simplify this concept for a layman, we can use integers and the $log_{10}$ method. For instance, $pmod{17}$ and $(-4)^2equiv -1$. Question: I’m feeling […]

## Is the square root of a number always greater than any of its prime factors?

Initially, I checked all prime numbers less than the integer $L$ to see if they were factors of $L$. Checking every prime less than $L$ becomes necessary if $L$ itself is a prime. Hence, to determine the minimum number of primes required to check before concluding that $L$ is prime, a brute force search is […]

## Calculating the quantity of whole number solutions to equations using the inclusion-exclusion principle

The number of non-negative integral solutions for positive integers $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$, with a maximum value of 5, is 365, as calculated by subtracting 350 from 715. Solution 2: The number of non-negative integer solutions for $X1+X2+…+XK=N$ is equivalent to the number of ways to distribute n identical balls into k labeled […]

## Expressing 99 as a multiplication of prime numbers

Solution 2 utilizes strong induction. The base case is when n equals 2, in which case n is prime and has factors of 1 and 2. The hypothesis assumes that for all k less than or equal to n, k is either prime or can be expressed as the product of prime factors. This hypothesis […]

## Prime Factorizations and the Addition of Two Perfect Squares

I understand that each factor of $2$ can be expressed as a sum of squares, specifically as $(1^2+1^2)$. However, I am unsure how to prove that all factors of the form $p_i^{a_i}$ and $q_j^{b_j}$ can also be expressed as sums of squares. It is important to note that if two numbers are written as the […]

## Equation of Pell in a General Form

In the method described by Prof. Lubin for the continued fraction of $sqrt{67} – 4$, the following equalities hold: $$sqrt { 13} – 3 = frac{ sqrt {13} – 3 }{ 1 } $$ $$sqrt { 13} – 1 = frac{ sqrt {13} – 1 }{4 } + 1 $$ $$sqrt { 13} – 2 […]

## Executing shots in billiards by launching at a 45 degree angle and rebounding off the boundaries

Each bounce in this scenario occurs at a multiple of $y$. The abscissa of the $n^{th}$ bounce is given by $yn bmod x$. The final bounce that lands the ball in the hole is when $yn bmod x=0$. The smallest value of $n$ that satisfies this condition is denoted as $n_h$, which can be calculated […]