The hottest Proofs Substack posts right now

Salinons are a geometric shape with a unique construction involving four semicircles.
Salinons have an interesting area formula that relates to a circle with the same area.
You can replace the semicircles in a salinon with other shapes, like rectangles or triangles, and still maintain certain area relationships.

Every positive integer can be split into a sum in a specific number of ways. For any integer n, there are exactly 2^(n-1) ways to do this.
To figure out how to split an integer, you can visualize it as a series of ones with spaces in between. Each space can either have a plus sign or not, giving rise to different sums.
A common mistake in math is the 'fence-post error,' where people confuse the number of items with the number of spaces between them, leading to miscounts in things like days or numbers.

Mathematical Induction is a technique for proving statements by starting with a base case and progressing through inductive steps. It forms the foundation for recursion.
Both Mathematical Induction and recursion rely on base cases, operate on discrete domains, and reduce problems to already proven statements. They are like mirror images of each other in problem-solving.
Understanding Mathematical Induction can greatly improve recursion skills as they share similar problem-solving approaches. Practicing PMI questions can enhance recursion proficiency.

Mathematical induction is a fundamental principle in mathematics, used to prove many fundamental facts in arithmetic and number theory.
The common induction principle states that if a set of natural numbers contains 0 and whenever n is in the set, n+1 is also in the set, then every natural number is in the set.
Strong induction allows the induction step to use multiple smaller numbers to prove a statement, and can be proven from the least-number principle.

Networks connect different objects like the internet, neurons in the brain, and roads in a city.
Graph theory studies connected elements in networks, using graphs with vertices and edges.
Graphs can show paths between vertices, connectivity in the network, and identify critical points called cut vertices.

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Mathematics requires proofs to truly understand concepts and ideas
Proofs in mathematics may seem like magic, but they are logical arguments based on ideas and patterns
The process of proving mathematical theorems involves clever ideas and logical reasoning

Mathematical proofs are essential for theoretical advancements in quantum information and mechanics.
Resources like the Lean and Coq projects offer tools for proof assistance and management.
Attending live sessions like 'Matrix Groups - Part 1' can offer valuable insights for enthusiasts.

Exercise 2.3 shows how to prove an equation with variables.
Exercise 2.4 explains demonstrating a series of equations with whole numbers.
Exercise 2.5 illustrates proving an equation involving different mathematical concepts.

Check for divisibility between two numbers by comparing their multiples with another common multiple, often using integers.
Composite integers have prime divisors that are less than or equal to the square root of the composite integer.
The number of multiples of a given integer within a specified interval can be calculated using floor functions and division.