Lagrange proved that for any natural number n that is not a perfect square there are x and y > 0 that satisfy Pell's equation.

This method always terminates with a solution ( proved by Lagrange in 1768 ).

Germain also proved or nearly proved several results that were attributed to Lagrange or were rediscovered years later.

These last four attempts assumed implicitly Girard's assertion ; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was a + bi for some real numbers a and b. In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial p ( z ).

The theorem was proved by Lagrange and generalized by Hans Heinrich Bürmann, both in the late 18th century.

Fermat, Euler, Lagrange, Legendre, and other number theorists of the 17th and 18th centuries proved some theorems and made some conjectures about quadratic residues, but the first systematic treatment is § IV of Gauss's Disquisitiones Arithmeticae ( 1801 ).

The case for squares, k = 2, was answered by Lagrange in 1770, who proved that every positive integer is the sum of at most four squares.

Joseph Louis Lagrange proved the square case in 1770, which states that every positive number can be represented as a sum of four squares, for example, .. Gauss proved the triangular case in 1796, commemorating the occasion by writing in his diary the line " ΕΥΡΗΚΑ!

If he did know this result it would be truly remarkable for even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Joseph Louis Lagrange proved it using results due to Leonhard Euler.

He had also intervened on behalf of a number of foreign-born scientists including mathematician Joseph Louis Lagrange, granting them exception to a mandate stripping all foreigners of possessions and freedom .< ref >

Lagrange did not prove his theorem ; all he did, essentially, was to discuss some special cases.

Within two years and a half he had mastered all the subjects prescribed for examination, and a great deal more, and, on going up for examination at Toulouse, he astounded his examiner by his knowledge of J. L. Lagrange.

Not all of the ring material would have necessarily been swept up right away ; the thickened crust of the Far Side suggests that a second moon about 1, 000-km in diameter formed in a Lagrange point of the Moon ; after tens of millions of years, as the two moons migrated outward from the Earth, solar tidal effects would have made the Lagrange orbit unstable, resulting in a slow-velocity collision that would have ' pancaked ' the smaller moon onto what is now the Far Side.

If a star grows outside of its Roche lobe too fast for all abundant matter to be transferred to the other component, it is also possible that matter will leave the system through other Lagrange points or as stellar wind, thus being effectively lost to both components.

We use Lagrange multipliers to find the point of maximum entropy,, across all discrete probability distributions on.

Dampier then followed the coast northeast, reaching the Dampier Archipelago and then Lagrange Bay, just south of what is now called Roebuck Bay all the while recording and collecting specimens, including many shells.

If on the one hand, this measure was thought as a response to workers ' alienation, on the other hand, the Popular Front gave Léo Lagrange ( SFIO ) responsibility for organisation of the use this leisure time, and of all aspects concerning sports.

His teachers recognized his talents in mathematics, but by 15 years of age he had already learned all the material taught at the school, and he began to study differential calculus from the works of Euler and Lagrange.

* Lagrange discusses representations of integers by general algebraic forms ; produces a tract on elimination theory ; publishes his first paper on the general process for solving an algebraic equation of any degree via Lagrange resolvents ; and proves Bachet's theorem that every positive integer is the sum of four squares.

Lagrange gave a proof in 1775 that was based on his study of quadratic forms.

Lagrange in 1773 initiated the development of the general theory of quadratic forms.

Lagrange gave a proof in 1770 based on his general theory of integral quadratic forms.

The discriminant of the quadratic form is defined to be ( this is the definition due to Gauss ; Lagrange did not require the term to have even coefficient, and defined the discriminant as ).

Sampling a polynomial of degree k − 1 at more than k points creates an overdetermined system, and allows recovery of the polynomial at the receiver given any k out of n sample points using ( Lagrange ) interpolation.

* Lagrange publishes his second paper on the general process for solving an algebraic equation of any degree via Lagrange resolvents ; and proves Wilson's theorem that if n is a prime, then ( n − 1 )!

Writing coordinates ( t, x ) = ( x < sup > 0 </ sup >, x < sup > 1 </ sup >, x < sup > 2 </ sup >, x < sup > 3 </ sup >) = x < sup > μ </ sup >, this form of the Euler – Lagrange equation is

The STScI is currently developing similar processes for JWST, although the operational details will be very different due to its different instrumentation and spacecraft constraints, and its location at the Sun-Earth L2 Lagrange point (~ 1. 5 million km from Earth ) rather than the low Earth orbit (~ 565 km ) used by HST.

On November 1, 1994, NASA launched the WIND spacecraft as a solar wind monitor to orbit the Earth's Lagrange point as the interplanetary component of the Global Geospace Science ( GGS ) Program within the International Solar Terrestrial Physics ( ISTP ) program.

Let and let be n + 1 distinct elements of K. Then for and by Lagrange interpolation we have.

Since the Lagrangian ( eq ( 1 )) contains second derivatives, the Euler – Lagrange equation of motion for this field is

Gaia will be launched on a Soyuz-FG rocket and will fly to the Sun – Earth Lagrange point L2 located approximately 1. 5 million kilometers from Earth.

The ACE robotic spacecraft was launched August 25, 1997 and is currently operating in a Lissajous orbit close to the L1 Lagrange point ( which lies between the Sun and the Earth at a distance of some 1. 5 million km from the latter ).

... being the Lagrange multipliers on the non-negativity constraints, the | last2 = Zhang | first2 = Shu Zhong | title = Pivot rules for linear programming: A Survey on recent theoretical developments | issue = Degeneracy in optimization problems | journal = Annals of Operations Research | volume = 46 – 47 | year = 1993 | issue = 1 | pages = 203 – 233 | doi = 10. 1007 / BF02096264 | mr = 1260019 | id = | publisher = Springer Netherlands | issn = 0254-5330 | ref = harv

After a successful thruster burn to knock it loose from its halo orbit on September 1 of that year, it used the instability of the Earth / Moon and Earth / Sun Lagrange points, making a series of lunar orbits over the next 15 months.

* Louis Alix, 2nd Duke of Cadore ( Saint-Vincent-de-Boisset, 11 January 1796-Boulogne-sur-Seine, 27 January 1870 ), married in Paris, 17 May 1824 Caroline Elisabeth de Lagrange ( Paris, 6 August 1806-Paris, 1 September 1870 )

The Lagrangian points (; also Lagrange points, L-points, or libration points ) are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be part of a constant-shape pattern with two larger objects ( such as a satellite with respect to the Earth and Moon ).

If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion ( known as the Lagrange or Euler – Lagrange equations ) are a set of equations:

These leading and trailing points are called the and Lagrange points.

Most inner moons of planets have synchronous rotation, so their synchronous orbits are, in practice, limited to their leading and trailing ( and ) Lagrange points, as well as the and Lagrange points, assuming they do not fall within the body of the moon.

Supporting this theory, extrasolar planets have been discovered in Lagrange points of each other, and are expected to collide, after co-orbiting for millions of years.

However, these linear orbits are not as stable as, for example, the equilateral Lagrange orbits and.

In classical mechanics, the equation of motion is Newton's second law, and equivalent formulations are the Euler – Lagrange equations and Hamilton's equations.

The Euler – Lagrange equations of motion for the functional E are then given in local coordinates by

Lagrange also reformulated the principles of classical mechanics, emphasizing energy more than force and developing a method to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic.

If is a maximum of for the original constrained problem, then there exists such that is a stationary point for the Lagrange function ( stationary points are those points where the partial derivatives of are zero, i. e. ).

are called Lagrange Multipliers and this optimization method itself is called The Method of Lagrange Multipliers.

In optimal control theory, the Lagrange multipliers are interpreted as costate variables, and Lagrange multipliers are reformulated as the minimization of the Hamiltonian, in Pontryagin's minimum principle.

The ITN makes particular use of Lagrange points as locations where trajectories through space are redirected using little or no energy.

In numerical analysis, Lagrange polynomials are used for polynomial interpolation.

Lagrange polynomials are used in the Newton – Cotes method of numerical integration and in Shamir's secret sharing scheme in cryptography.