* A simple explanation of Euler's Conjecture at Maths Is Good For You!

For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

For example, if one starts with Euler's totient function, and repeatedly applies the transformation process, one obtains:

For a semiprime n = pq the value of Euler's totient function ( the number of positive integers less than or equal to n that are relatively prime to n ) is particularly simple when p and q are distinct:

In the case of quantum mechanics, the main part of the three-body problem refers to the finding the eigenstates and their energies .. For a special case of the quantum three-body problem known as the Hydrogen Molecular ion, the eigenenergies are solvable analytically ( see discussion in quantum mechanical version of Euler's three-body problem ) in terms a generalization of the Lambert W function.

For Euler's totient function:

For instance, to explain the relationships and constants in Euler's identity by way of reference to motion and perception, e. g. pi as descriptive of the space swept out by an arm.

For example, using Cartesian coordinates on the plane, the distance between two points ( x < sub > 1 </ sub >, y < sub > 1 </ sub >) and ( x < sub > 2 </ sub >, y < sub > 2 </ sub >) is defined by the formula

For a first order predicate calculus, with no (" proper ") axioms, Gödel's completeness theorem states that the theorems ( provable statements ) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.

For example, primates have brains 5 to 10 times larger than the formula predicts.

For example, the formula a AND b is satisfiable because one can find the values a = TRUE and b = TRUE, which make ( a AND b ) = TRUE.

For example, if a graph has 17 valid 3-colorings, the SAT formula produced by the reduction will have 17 satisfying assignments.

For example, the formula for heavy water may be written D < sub > 2 </ sub > O instead of < sup > 2 </ sup > H < sub > 2 </ sub > O.

For instance, together with the spectral radius formula, it implies that the C *- norm is uniquely determined by the algebraic structure:

For example, Dalton assumed that water's formula was HO, giving the atomic weight of oxygen as eight times that of hydrogen, instead of the modern value of about 16.

For continuous cash flows, the summation in the above formula is replaced by an integration:

For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence.

For example the ester hexyl octanoate, also known under the trivial name hexyl caprylate, has the formula CH < sub > 3 </ sub >( CH < sub > 2 </ sub >)< sub > 6 </ sub > CO < sub > 2 </ sub >( CH < sub > 2 </ sub >)< sub > 5 </ sub > CH < sub > 3 </ sub >.

For example, the chemical compound n-hexane has the structural formula, which shows that it has 6 carbon atoms arranged in a chain, and 14 hydrogen atoms.

For example, formaldehyde, acetic acid and glucose have the same empirical formula,.

For example, in a system where there is no queuing, the GoS may be that no more than 1 call in 100 is blocked ( i. e., rejected ) due to all circuits being in use ( a GoS of 0. 01 ), which becomes the target probability of call blocking, P < sub > b </ sub >, when using the Erlang B formula.

For closed ( orientable or non-orientable ) surfaces with positive genus, the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula

For the diffusion equation this formula gives

For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras ; the Egyptians had a correct formula for the volume of a frustum of a square pyramid ;

For arbitrary n ≥ 2 we may generalize this formula, as noted above, by interpreting the third equation for the harmonic mean differently.

For instance, is not a well-formed formula, because we do not know if we are conjoining with or if we are conjoining with.

For the most part, they have the dental formula:

For example, if one tried to demonstrate it using the hydrocarbons decane ( chemical formula C < sub > 10 </ sub > H < sub > 22 </ sub >) and undecane ( C < sub > 11 </ sub > H < sub > 24 </ sub >), one would find that 100 grams of carbon could react with 18. 46 grams of hydrogen to produce decane or with 18. 31 grams of hydrogen to produce undecane, for a ratio of hydrogen masses of 121: 120, which is hardly a ratio of " small " whole numbers.

For example, the olivine group is described by the variable formula ( Mg, Fe )< sub > 2 </ sub > SiO < sub > 4 </ sub >, which is a solid solution of two end-member species, magnesium-rich forsterite and iron-rich fayalite, which are described by a fixed chemical formula.

For example, the field extension R / Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C / R and Q (√ 2 )/ Q are algebraic, where C is the field of complex numbers.

For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.

* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).

For example, if K is a field of characteristic p and if X is transcendental over K, is a non-separable algebraic field extension.

For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms.

Mesa has an " imperative " and " algebraic " syntax, in many respects more similar to ALGOL and Pascal than to C. For instance, compound commands are indicated by BEGIN / END keywords, rather than braces.

For example, rather than just getting a passing grade for mathematics, a student might be assessed as level 4 for number sense, level 5 for algebraic concepts, level 3 for measurement skills, etc.

For example, factorization or ramification of prime ideals when lifted to an extension field, a basic problem of algebraic number theory, bears some resemblance with ramification in geometry.

For example, the algebraic expression, if performed with a stack size of 4 and executed from left to right, would exhaust the stack.

For instance, let X be an algebraic variety with structure sheaf O < sub > X </ sub >.

For example, the familiar algebraic equation y = mx + b ( y

For example, algebraic topology makes use of Eilenberg – MacLane spaces which are spaces with prescribed homotopy groups.

For purposes of recording the moves that are played in the game, it is sufficient to employ an algebraic form of notation, as in Chess, and write the names of the pieces and the squares they are to be placed in.

For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms.

For instance, the corner square nearest white's left hand (" a1 " in algebraic notation ) is called " queen's rook 1 " ( QR1 ) by white and " queen's rook 8 " ( QR8 ) by black.

For example, Galois theory studies the connection between certain fields and groups, algebraic structures of two different kinds.

For another example, the group can be seen as a set that is equipped with an algebraic structure, namely the operation.

# Structures such as fields have some axioms that hold only for nonzero members of S. For an algebraic structure to be a variety, its operations must be defined for all members of S ; there can be no partial operations.

For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed.

For a few small values of and these rings of algebraic integers are PIDs, and this can be seen as an explanation of the classical successes of Fermat () and Euler ().

For instance the ring of ordinary integers is a PID, but as seen above the ring of algebraic integers in a number field need not be a PID.

For any fixed value of n these identities can be obtained by tedious but completely straightforward algebraic manipulations.