* James ` s Theorem For a Banach space the following two properties are equivalent:

For example, there are 20, 138, 200 Carmichael numbers between 1 and 10 < sup > 21 </ sup > ( approximately one in 50 billion numbers ).< ref name =" Pinch2007 "> Richard Pinch, " The Carmichael numbers up to 10 < sup > 21 </ sup >", May 2007 .</ ref > This makes tests based on Fermat's Little Theorem slightly risky compared to others such as the Solovay-Strassen primality test.

For larger values of n, Fermat's Last Theorem states there are no positive integer solutions ( x, y, z ).</ td >

Theorem 1. 6: For all elements in a group.

Theorem 1. 7: For all elements and in group,.

Theorem 1. 8: For all elements in a group, then.

Theorem 1. 3: For all elements in a group, there exists a unique such that, namely.

For with coprime and, one can use the Prime-Factor ( Good-Thomas ) algorithm ( PFA ), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley – Tukey but without the twiddle factors.

Theorem ( Modified Schwarz inequality for 2-positive maps ) For a 2-positive map Φ between C *- algebras, for all a, b in its domain,

For any flow, you can write the equations of the flow in terms of vorticity rather than velocity by simply taking the curl of the flow equations that are framed in terms of velocity ( may have to apply the 2nd Fundamental Theorem of Calculus to do this rigorously ).

For example Newton wrote to Oldenberg in 1676 " I believe M. Leibnitz will not dislike the Theorem towards the beginning of my letter pag.

For dimension 2, there is a rich structure in virtue of the absence of Desargues ' Theorem.

For example, when students are discovering the formula for the Pythagorean Theorem in math class, the teacher may identify hints or cues to help the student reach an even higher level of thinking.

( For example, the means of sufficiently large samples drawn from practically any underlying distribution whose variance exists are normally distributed, according to the Central Limit Theorem.

</ li >< li > Pythagorean Theorem: For and meeting on orthogonal lines at ( )

Theorem For ƒ in L < sup > 1 </ sup >( R < sup > d </ sup >), the above series converges pointwise almost everywhere, and thus defines a periodic function Pƒ on Λ. Pƒ lies in L < sup > 1 </ sup >( Λ ) with || Pƒ ||< sub > 1 </ sub > ≤ || ƒ ||< sub > 1 </ sub >.

Theorem For a bounded operator T, σ < sub > r </ sub >( T ) ⊂ σ < sub > p </ sub >( T *) ⊂ σ < sub > r </ sub >( T ) ∪ σ < sub > p </ sub >( T ).

His Ph. D. dissertation, written under the direction of Richard Courant and Bernard Friedman at New York University, was on the topic " The Bridge Theorem For Minimal Surfaces.

: Theorem 1: For any pair of algorithms a < sub > 1 </ sub > and a < sub > 2 </ sub >

For such objects, the integral may be taken over the entire surface () by taking the absolute value of the integrand ( so that the " top " and " bottom " of the object do not subtract away, as would be required by the Divergence Theorem applied to the constant vector field ) and dividing by two:

Theorem For any normalized continuous positive definite function f on G ( normalization here means f is 1 at the unit of G ), there exists a unique probability measure on such that

For a proof of the first part of the Latin square property, see the Wikipedia page on elementary group theory ( Theorem 1. 3 ).

For although I had crossed a corner of the hall on my way to the toilet I still could not tell for sure how far to the rear the darkness extended.

For weeks I wandered about this neighborhood of warehouses and garages, truck terminals and taxi repair shops, gasoline pumps and longshoremen's lunch counters, yet never did I cease to feel myself a stranger there.

Later Helion wrote of this phase: `` For years I built for myself a subtle instrument of relationships -- colors and forms without a name.

For pride's sake, I will not say that the coy and leering vade mecum of those verses insinuated itself into my soul.

For innocence, of all the graces of the spirit, is I believe the one most to be prayed for.

For a delightful drive out of Athens I should recommend Sounion, at the end of the Attic Peninsula.

For this reason, then I want to describe, first, two examples of the puritanical attacks: Stephen Gosson's The School Of Abuse, 1579, and his later Playes Confuted, published in 1582.

For my part I find it difficult to conceive such a state of affairs.

For a while there was such shrill girlish commotion I couldn't have made myself heard if I'd had the equivalent of the message to Garcia.

For, after leaving the Army in 1956, I spent five years in Graduate School first at Boston College and then at the University of Toronto.

`` For a time I thought of trying to reach the Free Polish Forces, but one thing led to another.

For the near term, however, it must be realized that the industrial and commercial market is somewhat more sensitive to general business conditions than is the military market, and for this reason I would expect that any gain in 1961 may be somewhat smaller than those of recent years ; ;

For my seedbed I use good garden soil with a little sand added to encourage rooting.

For added strength, I also fastened a small block on each side of every frame and batten joint.

For padding the seats and bunks, I used Ensolite, Type Aj.

For me, these will belong more completely to their surroundings if they are conceived in this early stage, though I freely admit that I do not hesitate to add or eliminate figures on the full sheet when it serves my final purpose.

For example, one hebephrenic man used to annoy me, month after month, by saying, whenever I got up to leave and made my fairly steoreotyped comment that I would be seeing him on the following day, or whenever, `` You're welcome '', in a notably condescending fashion -- as though it were his due for me to thank him for the privilege of spending the hour with him, and he were thus pointing up my failure to utter a humbly grateful, `` thank you '' to him at the end of each session.

For Yokuts, I tabulated these 71 items in five columns, according as they were expressed by 1, 2, 3, 4, and more than 4 stems.

For the reason just suggested, I shall assume the use of the first subtype of fully distributed cost apportionment in the following simplified example.

For what I express in my remark is something going on in me at the time, and that of course did not exist until I did come on the scene.