Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π ( x ) were strong enough for him to prove Bertrand's postulate that there exists a prime number between n and 2n for any integer n ≥ 2.

Although Kummer had studied Fermat's Last Theorem in the 1830s and was probably aware that his theory would have implications for its study, it is more likely that the subject of Jacobi's ( and Gauss's ) interest, higher reciprocity laws, held more importance for him.

Although debate over how to interpret this artifact continues, the artifact is usually taken to display knowledge of the Pythagorean Theorem, long before the birth of Pythagoras himself.

* Although Sard's Theorem does not hold in general, every continuous map f: X → R < sup > n </ sup > from a Hilbert manifold can be arbitrary closely approximated by a smooth map g: X → R < sup > n </ sup > which has no critical points

Although he questions the extent and nature of the alleged revival of religion and the alleged increase in conformity, and thinks that `` hedonistic '' present-time orientation does not have the meaning usually attributed to it, he does conclude that Americans increasingly enjoy leisure without guilt, do not stress achievement so much as formerly, are more accepting of group harmony as a goal, more tolerant of diversity and aware of other cultures.

Although Romans who had been adopted into a new family usually retained their old nomen in cognomen form ( e. g. Octavianus for one who had been an Octavius, Aemilianus for one who had been an Aemilius, etc.

Although works of precious gems and metals usually do not survive the ravages of time, some fine pieces of Assyrian jewelry were found in royal tombs at Nimrud.

Although variations of the blood supply to the adrenal glands ( and indeed the kidneys themselves ) are common, there are usually three arteries that supply each adrenal gland:

Although players usually use their sticks to move the ball around, they may use any part of their bodies other than their heads, hands or arms and may use their skates in a limited manner.

Although nude men were depicted standing upright, most depictions of female nudity in Greek art occurred " usually with drapery near at hand and with a forward-bending, self-protecting posture ".

Although not always seen, her mourning call is heard, usually at night when someone is about to die and usually around woods.

Although Douglas was always second billed under Lancaster in these films, their roles were usually more or less the same size with the exceptions of I Walk Alone, in which Douglas played a villain, and in Seven Days in May, where Douglas ' part was larger than Lancaster's but not as dramatic.

Although these two are often considered interchangeable, compilers and execution environments usually assign different signals to these.

Although he is usually credited with coining the phrase, he always insisted that it was suggested to him by somebody else.

Although cadmium usually has an oxidation state of + 2, it also exists in the + 1 state.

Although organizational communication, as a field of study, is usually geared toward companies and business groups, these may also be seen as communities.

Although the best treatment for comatose patients remains unknown, hospitals usually place comatose patients in an Intensive Care Unit ( ICU ) immediately.

Although one of his brothers became a clergyman and one of his sisters a missionary, Attlee himself is usually regarded as an agnostic.

Although all group 16 elements of the periodic table, including oxygen, are defined as chalcogens, oxygen and oxides are usually distinguished from chalcogens and chalcogenides.

Although it is usually translated as " element ", the Chinese word xing literally means something like " changing states of being ", " permutations " or " metamorphoses of being ".

( Although mutual orthogonality is the only condition, these vectors are usually constructed for ease of decoding, for example columns or rows from Walsh matrices.

Although production was usually about 57, 000 – 59, 000 tons annually in the 1980s, the acreage harvested declined from in the early 1980s to in 1999, indicating a greater yield per acre.

Although The Moonstone is usually seen as the first detective novel, a number of critics suggest that the lesser known Notting Hill Mystery ( 1862 – 63 ), written by the pseudonymous " Charles Felix ", preceded it by a number of years and first used techniques that would come to define the genre.

Although Tacitus is usually considered to be the most reliable author of this era, his views on Domitian are complicated by the fact that his father-in-law, Gnaeus Julius Agricola, may have been a personal enemy of the Emperor.

Although not usually stated in polynomial form, this example is equivalent to the polynomial equation 4xyz = yzn + xzn + xyn = n ( yz + xz + xy ).</ td >

Although usually trivalent, europium readily forms divalent compounds.

Although sunlight is free and abundant, solar electricity is still usually more expensive to produce than large-scale mechanically generated power due to the cost of the panels.

Although ephemeris time was defined in principle by the orbital motion of the Earth around the Sun, it was usually measured in practice by the orbital motion of the Moon around the Earth.

Although " android " is used almost universally to refer to both sexes, and those of no particular sex, " android " technically refers to the male form, while " gynoid " is the feminine form.

Although MND refers to a specific subset of pathologically similar diseases ; there are numerous other afflictions of motor neurons that are pathologically distinct from MND and have a different clinical course.

Although Muslim scholars have often debated over who this verse refers to, most classical commentators and modern translators have taken the opinion that this refers to Amram, the father of Moses and Aaron.

Although these churches hold much in common, their theology differs on whether the intermediate state is unconscious sleep or consciousness, whether the ultimate punishment of the wicked is annihilation or eternal torment, the nature of immortality, whether or not the wicked are resurrected after the millennium, and whether the sanctuary of refers to the one in heaven or one on earth.

" Upham's book refers to Robert Calef 25 times with the majority of these regarding documents compiled by Calef in the mid-1690s and stating: " Although zealously devoted to the work of exposing the enormities connected with the witchcraft prosecutions, there is no ground to dispute the veracity of Calef as to matters of fact.

Although The Porisms is lost, we know three lemmas contained there, since Diophantus refers to them in the Arithmetica.

Although the words faith and belief are sometimes erroneously conflated and used as synonyms, faith properly refers to a particular type ( or subset ) of belief, as defined above.

Although the Greek-language-origin prefix geo refers to Earth, " geology " is often used in conjunction with the names of other planetary bodies when describing their composition and internal processes: examples are " the geology of Mars " and " Lunar geology ".

Although imperialist practices have existed for thousands of years, the term " Age of Imperialism " generally refers to the activities of nations such as the United Kingdom, France, Germany, Italy, Japan and the United States in the early 18th through the middle 20th centuries, e. g., the " The Great Game " in Persian lands, the " Scramble for Africa " and the " Open Door Policy " in China.

Although the word jackal has been historically used to refer to many small-to medium-sized species of the wolf genus of mammals, Canis, today it most properly and commonly refers to three species: the black-backed jackal and the side-striped jackal of sub-Saharan Africa, and the golden jackal of northern Africa and south-central Eurasia.

Although the term lycanthropy properly speaking refers to metamorphosis into a wolf ( see werewolf ), lycanthropy is in popular practice used of transformation into any animal, even though the proper term is therianthropy.

Although Plato famously condemned poetic myth when discussing the education of the young in the Republic, primarily on the grounds that there was a danger that the young and uneducated might take the stories of Gods and heroes literally, nevertheless he constantly refers to myths of all kinds throughout his writings.

"( ) Although the term is not specifically used of her in the New Testament, Eastern Christianity refers to her as " Equal to the Apostles "), and later traditions name her as " the apostle to the apostles.

Although drying may refer to evaporation of the solvent or thinner, it usually refers to oxidative cross-linking of the binders and is indistinguishable from curing.

Although energy generated by stimulated emission is always at the exact frequency of the field which has stimulated it, the above rate equation refers only to excitation at the particular optical frequency corresponding to the energy of the transition.

Although " terrorism " originally referred to acts committed by a government, currently it usually refers to the killing of innocent people by a non-government group in such a way as to create a media spectacle.

Although the most common colloquial use of the word " collision " refers to accidents in which two or more objects collide, the scientific use of the word " collision " implies nothing about the magnitude of the forces.

Although journal is sometimes used as a synonym for " magazine ", in academic use, a journal refers to a serious, scholarly publication that is peer-reviewed.

Although it generally refers to religious beliefs that are accepted regardless of evidence, they can refer to acceptable opinions of philosophers or philosophical schools, public decrees, or issued decisions of political authorities .< ref >, " Dogma " The Concise Oxford Dictionary of World Religions.

Although changes of quantum state occur on the submicroscopic level, in popular discourse, the term " quantum leap " refers to a large increase.

Although the word flange generally refers to the actual raised rim or lip of a fitting, many flanged plumbing fittings are themselves known as ' flanges ':

Although painterly generally refers to a certain use of paint in art, it happens that some forms of sculpture make use of apparently random surface effects which, if not exactly resembling brushstrokes, contain the traits of painterliness ( see Wood as a medium ).

Although the term commonly refers to wireless data transfer mechanisms ( e. g. using radio, hypersonic or infrared systems ), it also encompasses data transferred over other media such as a telephone or computer network, optical link or other wired communications like phase line carriers.

Although several other groups of crustaceans have the word " lobster " in their names, the unqualified term " lobster " generally refers to the clawed lobsters of the family Nephropidae.