Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite.

Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.

* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.

# Every prime ideal of A is principal.

Every twin prime pair except ( 3, 5 ) is of the form ( 6n − 1, 6n + 1 ) for some natural number n, and with the exception of < var > n </ var > = 1, < var > n </ var > must end in 0, 2, 3, 5, 7, or 8.

Every third odd number is divisible by 3, which requires that no three successive odd numbers can be prime unless one of them is 3.

* Hardy and Littlewood listed as their Conjecture I: " Every large odd number ( n > 5 ) is the sum of a prime and the double of a prime.

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

Every strong probable prime to base a is also an Euler probable prime to the same base, but not vice versa.

Every closed 3-manifold has a prime decomposition: this means it is the connected sum of prime three-manifolds ( this decomposition is essentially unique except for a small problem in the case of non-orientable manifolds ).

: Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.

Every non-zero characteristic is a prime number.

: Every Boolean algebra contains a prime ideal.

* Every number of the form 2 < sup > m </ sup > p for a natural number m and a prime number p such that p < 2 < sup > m + 1 </ sup > is also semiperfect.

Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by

Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form, so by Galois theory, a-extension is the same thing as a tower of fields such that.

Every imperfect field is necessarily transcendental over its prime subfield ( the minimal subfield ), because the latter is perfect.

A finitely-generated abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form for some prime number p and some positive integer n. Every finitely-generated abelian group is a direct sum of ( finitely many ) indecomposable abelian groups.

* Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco.

Every entire function can be represented as a power series that converges uniformly on compact sets.

* Every soul is potentially divine, with innate qualities of infinite knowledge, perception, power, and bliss ( masked by its karmas ).

The newest formulas of Mobilgas Special was advertised as offering " A Tune-Up in Every Tankful " due to a combination of chemicals known as the " Mobil Power Compound " which was designed to increase power, check pre-ignition ping, correct spark plug misfiring, control stalling and combat gumming up of carburetors.

* k = 1 ( criticality ): Every fission causes an average of one more fission, leading to a fission ( and power ) level that is constant.

Every member would have decision-making power in the firm and would be able to participate in establishing its overall policy objectives.

Every aspect of life, every word, plant, animal and ritual was connected to the power and authority of the gods.

Every car uses either a 5. 0 L Ford " Boss 302 " SVO or a 5. 0 L Chevrolet small block race-engine ( depending on the make )-capable of producing between 460 and 485 kW ( 620 — 650 bhp ) of power, but generally quoted as a little over 450 kW ( 600 bhp ) in race trim.

Additional charges were assessed as follows: Every province receiving power from the Three Gorges Dam has to pay ¥ 7. 00 per MWh extra.

Every sailing ship has a hull, rigging and at least one mast to hold up the sails that use the wind to power the ship.

Every day between 300 and 1, 000 external employees are working in the power plant.

Every person going to Tribal Council had to compete, and the winner of the challenge won an additional power at Tribal Council.

Every aspect of life, every word, plant, animal and ritual was connected to the power and authority of the gods.

* Every power ought to be commensurate with its object.

“ Every machine is constructed for the purpose of performing certain mechanical operations, each of which supposes the existence of two other things besides the machine in question, namely, a moving power, and an object subject to the operation, which may be termed the work to be done.

Hitler even quoted the Dearborn Independent in Mein Kampf and Henry Ford was the only American that Hitler specifically named: " Every year they Jews manage to become increasingly the controlling masters of the labor power of a people of 120, 000, 000 souls ; one great man, Ford, to their exasperation still holds out independently there even now.

Every power supply must obtain the energy it supplies to its load, as well as any energy it consumes while performing that task, from an energy source.

In one statement he said: " Every virile people has established colonial power.

" A man's home is his castle " is translated in Gor as, " Every man is an Ubar within the circle of his sword " ( The Ubar is a war-leader, a General who takes power at a time of crisis, and whose rule is tantamount to tyrant until the crisis is resolved.

* Every practical number that is not a power of two is semiperfect.

Every successive Communist government throughout the world has followed this same pattern of first promising indigenous national minorities self-determination in order to gain their support and then actively opposing that self-determination once in power.

Every Polish town was bounded to put up a quantity of soldiers-this was a conspicuous sign of a power of a given town how much soldiers it had to put up.

Every major power on the planet goes on missile alert, ordering retaliatory strikes as soon as the missile warhead explodes on impact.

Condon himself once said: " Every book I've ever written has been about abuse of power.

Every child born in Fiji on or after the date on which the Constitution took effect is deemed to be a citizen of Fiji, provided that at least one parent is a citizen, and that neither parent is an accredited diplomat of a foreign power.

Every time they fight, Dark Specter is drained of power and grows weaker.