Analysis can assist number theory {analytic number theory}.
Rational numbers define cuts {Dedekind cut} in rational-number ordered set, so left side is less than or equal to rational number and right side is greater than rational number. Real numbers are limits of Dedekind-cut convergent sequences. Limits are rational-approximation converging-sequence limits. Irrational numbers partition rational-number sets.
For integer n > 2, x1^(n + 1) + x2^(n + 1) + ... + xn^(n + 1) = z^(n + 1), where z and x1 through xn are integers, has no positive integer solutions {Euler conjecture}. Case x1^4 + x2^4 + x3^4 = z^4 (n + 1 = 4) is false. Case x1^5 + x2^5 + x3^5 + x4^5 = z^5 (n + 1 = 5) is false. Fermat's last theorem is case x1^3 + x2^3 = z^3, where n = 2 (n + 1 = 3), which is true.
For integer w > 2, x^w + y^w = z^w, where x y z are integers, has no positive integer solutions {Fermat's last theorem} {Fermat last theorem}. Fermat proved that x^3 + y^3 = z^3 has no positive integer solutions. Andrew Wiles proved theorem for all cases [1993].
properties
x^2 + y^2 = z^2 has solutions for x = 3, 4, 5, and so on. z is odd, x is odd, and y is even. x + y > z. In parameterization, x = 2*m*n, y = m^2 - n^2, and z = m^2 + n^2.
Multiples of Pythagorean triples are Pythagorean triples. For lowest Pythagorean triples, z - y = 1 if x is odd, or z - y = 2 if x is even. For lowest Pythagorean triples, n = 1, if lowest of x or y is even, or m - n = 1, if lowest of x or y is odd.
x^2 + y^2 = z^2 has (x^2)/(z^2) + (y^2)/(z^2) = 1 and ((x^2 + y^2)^0.5)/z = 1, so one percent plus other percent = 100%. x^2/z^2 + y^2/z^2 + 2*x*y/z^2 = 1 + 2*x*y/z^2 means (x + y)^2 = z^2 + 2*x*y, where (x + y)^2 is area of square whose side is straight line of x + y, and 2*x*y is two times area of triangle rectangle.
triangle
For x + y = z, three natural numbers lie on a straight line, with xy angle 180 degrees. For x^2 + y^2 = z^2, three natural numbers lie on a right triangle, with xy angle 90 degrees. Perhaps, for x^3 + y^3 = z^3, three natural numbers lie on a triangle with xy angle 60 degrees, but this only allows x = y = z, so no natural number solutions. Perhaps, for x^4 + y^4 = z^4, three natural numbers lie on a triangle with xy angle 45 degrees, but this only allows x < x and z > y, so no natural number solutions.
cube
x^3 + y^3 = z^3 makes (x + y)^3 = z^3 + 3*x^2*y + 3*x*y^2. 3 * x^2 * y + 3 * x * y^2 = 3 * x * y * (x + y). (x + y)^3 is volume of cube with side x + y. 3 * x * y * (x + y) is three times volume of rectilinear solid with sides x, y, and x + y. x + y > z. Perhaps, side lengths and angles make an impossible figure. Perhaps, all higher powers make impossible figures. Perhaps, x^3 + y^3 = z^3 requires not odd and even properties, but three-part system with divisible-by-3 numbers, numbers one higher, and numbers two higher. x, y, and z must come from different categories. Perhaps, this is impossible.
x^2 - A * y^2 = 1 {Pell's equation} {Pell equation}, where A is integer.
Rules for operations on integers can be for all algebras {permanence of form}.
One and only one number adds or multiplies number to give another number {uniqueness law, number}.
Integers are sums of at most nine cubes {Waring's theorem} {Waring theorem}.
If number is greater than zero, smaller number added to itself enough times can equal the number {axiom of Archimedes, number} {Archimedes axiom, number}.
Number-theory axioms {calculation axiom} can be about calculation, such as associative law, commutative law, and distributive law.
Adding anything to real numbers makes all preceding axioms untrue, so real-number system cannot be larger {axiom of completeness} {completeness axiom}.
Number-theory axioms {connection axiom} can be about operations, such as closure, uniqueness, and identity.
Number-theory axioms {continuity axiom} can be about continuity, such as axiom of Archimedes and axiom of completeness.
Number-theory axioms {order axiom} can be about order, such as transitive law. For two different numbers, one number is greater and one number is smaller. If first number is greater than second number, then first number plus third number is greater than second number plus third number. If first number is greater than second number, then first number times third number is greater than second number times third number.
If n > 7, n! ~ n^n * e^-n * (2 * pi * n)^0.5, where n is integer, and e is base of natural logarithms {Stirling's theorem} {Stirling theorem} {Stirling's formula}.
Numbers equal to (p - 1)! + 1 are divisible by p if and only if p is prime {Wilson's theorem} {Wilson theorem}.
Rational numbers have order {order relation}.
If a is less than b, then a + c is less than b + c for all c, and a*c is less than b*c for all positive c {consistency relation}, where a b c are rational numbers.
If a is less than b, some c is greater than a and less than b {density relation}, where a b c are rational numbers.
For interval from b to c, some a are less than c and greater than b {extension relation}, where a b c are rational numbers.
If a is less than b, and if b is less than c, then a is less than c {transitivity relation} {transitive law} {transitivity, number}, where a b c are rational numbers.
For rational numbers a and b, a is greater than b, equal to b, or less than b {trichotomy relation}.
Ratio between number of prime numbers less than or equal to an integer and integer can have an approximation {prime number theorem}. Number of primes not exceeding number n is PI(n), whose limit can find the prime numbers: limit of PI(n) / (n / log(n)) = 1, as n goes to infinity.
Number of primes is infinite {Euclid's theorem} {Euclid theorem}.
If p is prime, and a is an integer with no common factor with p, then a^(p - 1) / p has remainder one {Fermat's theorem} {Fermat theorem}.
Positive even integers are sums of two primes {Goldbach's hypothesis} {Goldbach hypothesis} {Goldbach's conjecture}.
Odd numbers can be primes plus two times primes {Levy's conjecture} {Levy conjecture}: p' + 2 * q' = 2*n + 1, where n goes from 0 to infinity.
Algorithms {Shor's algorithm} {Shor algorithm} can find prime factors.
modular
Modular arithmetics have circular sets of numbers. Mathematical operations are periodic.
process
Start with mod. Using any number smaller than the mod, take its first, second, and so on, powers and express result in the mod until number sequence shows a repeating pattern. Distance between repeats is period. Divide period by two and use result as mod exponent. If period divided by two is not even number, start over.
factors
Take the integers one above and one below result. Find largest common divisor of number and two integers to calculate number factors.
From natural-number list, cross out all second numbers except for number two, then cross out all third numbers except for number three, and so on {sieve of Eratosthenes} {Eratosthenes sieve}. What remains are prime numbers.
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Date Modified: 2022.0225