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Theorem on the number of primes in arithmetic sequences
an arithmetic progression a , a + d , a + 2 d , a + 3 d , … , {\displaystyle a,\ a+d,\ a+2d,\ a+3d,\ \dots ,\ } and Dirichlet's theorem states
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
Concept in number theory
Dirichlet's theorem on arithmetic progressions Hurwitz's theorem (number theory) Heilbronn set Kronecker's theorem (generalization of Dirichlet's theorem)
Dirichlet's approximation theorem
Dirichlet's_approximation_theorem
Set of prime numbers linked by a linear relationship
an+b} , where a and b are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely
Primes in arithmetic progression
Primes_in_arithmetic_progression
Topics referred to by the same term
Dirichlet's theorem may refer to any of several mathematical theorems due to Peter Gustav Lejeune Dirichlet. Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem
Theorem about prime numbers
Green–Tao theorem, proven by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In
Green–Tao_theorem
Characterization of how many integers are prime
modulo d with gcd(a, d) = 1 . This is stronger than Dirichlet's theorem on arithmetic progressions (which only states that there is an infinity of primes
Prime_number_theorem
Exploring properties of the integers with complex analysis
Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well
Analytic_number_theory
Infinitely many prime numbers exist
proof, by the Swiss mathematician Leonhard Euler, relies on the fundamental theorem of arithmetic: that every integer has a unique prime factorization. What
Euclid's_theorem
Subset of mathematical connundrums
Green–Tao theorem. See also Dirichlet's theorem on arithmetic progressions. As of 2020[update], the longest known arithmetic progression of primes has length
Problems involving arithmetic progressions
Problems_involving_arithmetic_progressions
Says when a natural number is the sum of three squares of integers
to Dirichlet (in 1850), and has become classical. It requires three main lemmas: the quadratic reciprocity law, Dirichlet's theorem on arithmetic progressions
Legendre's three-square theorem
Legendre's_three-square_theorem
Number divisible only by 1 and itself
relatively prime, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The Green–Tao theorem shows that
Prime_number
Describes statistically the splitting of primes in a given Galois extension of Q
viewed as a generalisation of Dirichlet's theorem on arithmetic progressions. A quantitative form of Dirichlet's theorem states that if n ≥ 2 {\displaystyle
Chebotarev_density_theorem
Branch of pure mathematics
conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions. He gave a full treatment of the equation a x
Number_theory
Brun–Titchmarsh theorem, named after Viggo Brun and Edward Charles Titchmarsh, is an upper bound on the distribution of prime numbers in arithmetic progression. Let
Brun–Titchmarsh_theorem
Number of integers coprime to and less than n
which itself is a corollary of the proof of Dirichlet's theorem on arithmetic progressions. The Dirichlet series for φ(n) may be written in terms of the
Euler's_totient_function
Mathematical theorem
Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that there exist
Linnik's_theorem
Bruijn–Newman constant Dirichlet character Dirichlet L-series Siegel zero Dirichlet's theorem on arithmetic progressions Linnik's theorem Elliott–Halberstam
List_of_number_theory_topics
German mathematician (1805–1859)
of which were later named after him. In 1837, Dirichlet proved his theorem on arithmetic progressions using concepts from mathematical analysis to tackle
Peter Gustav Lejeune Dirichlet
Peter_Gustav_Lejeune_Dirichlet
Concept in number theory
example, in proving Dirichlet's theorem on arithmetic progressions, it is easy to show that the set of primes in an arithmetic progression a + nb (for a, b
Dirichlet_density
approximations) Dirichlet's theorem on arithmetic progressions (number theory) Dirichlet's unit theorem (algebraic number theory) Equidistribution theorem (ergodic
List_of_theorems
On prime divisors of differences two nth powers
theorem Wilson prime Kaprekar's constant Fermat's little theorem Palindromic numbers Harshad numbers Dirichlet's theorem on arithmetic progressions A
Zsigmondy's_theorem
a refinement both of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions. Define ψ ( x ; q , a ) = ∑ n ≤ x n ≡ a
Siegel–Walfisz_theorem
Number representing illegal information
would produce the same decompression. This echoes the Dirichlet's theorem on arithmetic progressions, where it is proven that for coprime integers b and
Illegal_number
Branch of mathematics
classes are generalised arithmetic progressions or generalised ideal classes. If χ is a character of A then we can define a Dirichlet series ∑ g ∈ G χ ( [
Abstract analytic number theory
Abstract_analytic_number_theory
Diophantine equation Dirichlet 1. Dirichlet's theorem on arithmetic progressions 2. Dirichlet character 3. Dirichlet's unit theorem. distribution A distribution
Glossary_of_number_theory
Conjecture on zeros of the zeta function
an explicit version of a theorem of Cramér. The Riemann hypothesis implies strong bounds on the growth of many other arithmetic functions, in addition to
Riemann_hypothesis
Formula whose values are the prime numbers
{\displaystyle n} ranging from -42 to 15. It is known, based on Dirichlet's theorem on arithmetic progressions, that linear polynomial functions L ( n ) = a n +
Formula_for_primes
Mathematical conjecture about zeros of L-functions
ordinary Riemann hypothesis. More effective version of Dirichlet's theorem on arithmetic progressions: Let π ( x , a , d ) {\textstyle \pi (x,a,d)} where
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Complex-valued arithmetic function
who introduced these functions in his 1837 paper on primes in arithmetic progressions. They are a prominent example of the general idea of a character
Dirichlet_character
Type of mathematical function
who introduced them in 1837 to prove his theorem on primes in arithmetic progressions. In his proof, Dirichlet showed that L ( s , χ ) {\displaystyle L(s
Dirichlet_L-function
(1948), "On arithmetical properties of Lambert series" (PDF), J. Indian Math. Soc., New Series, 12: 63–66, MR 0029405 Borwein, Peter B. (1992), "On the irrationality
List_of_numbers
Meromorphic function on the complex plane
7, Section 2, 1992, p. 455. P. G. L. Dirichlet: Proof of the theorem that every infinite arithmetic progression whose first term and common difference
L-function
Mathematical theorem
obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over a range of moduli. The first result of this kind
Bombieri–Vinogradov_theorem
Australian and American mathematician (born 1975)
source of Green and Tao's arithmetic progressions is Endre Szemerédi's 1975 theorem on existence of arithmetic progressions in certain sets of integers
Terence_Tao
(diophantine approximation) Dirichlet's theorem on arithmetic progressions (number theory, specifically prime numbers) Dirichlet's unit theorem (algebraic number
List of things named after Peter Gustav Lejeune Dirichlet
List_of_things_named_after_Peter_Gustav_Lejeune_Dirichlet
Mathematical term; concerning axioms used to derive theorems
OCLC 429049174. Zalta, Edward N. (Aug 5, 2023). "Frege's Theorem and Foundations for Arithmetic". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy
Axiomatic_system
Basic result of approximation theory
There are also versions for the Lp spaces. Erdős conjecture on arithmetic progressions Müntz, Ch. H. (1914). "Über den Approximationssatz von Weierstrass"
Müntz–Szász_theorem
Series of four mathematics textbooks
presents applications to partial differential equations, Dirichlet's theorem on arithmetic progressions, and other topics. Because Lebesgue integration is not
Princeton Lectures in Analysis
Princeton_Lectures_in_Analysis
their L-functions to establish Dirichlet's theorem on arithmetic progressions. In subsequent publications, Dirichlet used these tools to determine, among
List of publications in mathematics
List_of_publications_in_mathematics
Irrational number based on primes
be proven with Dirichlet's theorem on arithmetic progressions or Bertrand's postulate (Hardy and Wright, p. 113) or Ramare's theorem that every even
Copeland–Erdős_constant
Number that is the result of operation on its own digits
b+1} are always relatively prime, and therefore, by Dirichlet's theorem on arithmetic progressions, the sequence contains an infinite number of primes
Friedman_number
Provides an asymptotic formula for counting the number of prime ideals of a number field
primes in the arithmetic progression 4n + 1, and r′ in the arithmetic progression 4n + 3. By the quantitative form of Dirichlet's theorem on primes, each
Landau_prime_ideal_theorem
Technique in analytic number theory by Helmut Maier
and the columns are arithmetic progressions where the difference is the primorial. By Dirichlet's theorem on arithmetic progressions the columns will contain
Maier's_matrix_method
modulo an integer is an area of active research. Dirichlet's theorem on primes in arithmetic progressions shows that there are an infinity of primes in each
Multiplicative_number_theory
Set of integers whose sum of reciprocals diverges
all primes in an arithmetic progression an + b, where a and b are coprime is large (see Dirichlet's theorem on arithmetic progressions). Every subset of
Large_set_(combinatorics)
Number theory conjecture
needed] The special case of a single linear polynomial is Dirichlet's theorem on arithmetic progressions, one of the most important results of number theory
Schinzel's_hypothesis_H
k is a unit. According to Dirichlet's theorem on arithmetic progressions there are infinitely many primes in the progression, and for a prime p {\displaystyle
Root_of_unity_modulo_n
Argand in 1806. Dirichlet's theorem on arithmetic progressions. In 1808 Legendre published an attempt at a proof of Dirichlet's theorem, but as Dupré pointed
List_of_incomplete_proofs
American mathematician
combined Bertrand's postulate with Dirichlet's theorem on arithmetic progressions by showing that each of the progressions 3i + 1, 3i + 2, 4i + 1, and 4i + 3
Robert_Breusch
Irreducible polynomial whose roots are nth roots of unity
congruent to 1 modulo n, which is a special case of Dirichlet's theorem on arithmetic progressions. The constant-coefficient linear recurrences which are
Cyclotomic_polynomial
Origin and evolution of the symbols used to write equations and formulas
Peter Gustav Lejeune Dirichlet developed Dirichlet L-functions to give the proof of Dirichlet's theorem on arithmetic progressions and began analytic number
History of mathematical notation
History_of_mathematical_notation
Gives conditions for the solvability of quadratic equations modulo prime numbers
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations
Quadratic_reciprocity
\mathbb {Q} (i)} . 1837 Dirichlet's theorem on arithmetic progressions. 1853 Leopold Kronecker announces the Kronecker–Weber theorem 1880 Kronecker introduces
Timeline of class field theory
Timeline_of_class_field_theory
Prime number congruent to 1 mod 4
101, 109, 113, ... (sequence A002144 in the OEIS). By Dirichlet's theorem on arithmetic progressions, this sequence is infinite. More strongly, for each
Pythagorean_prime
Lejeune Dirichlet proves Fermat's Last Theorem for n = 14. 1835 – Lejeune Dirichlet proves Dirichlet's theorem about prime numbers in arithmetical progressions
Timeline_of_mathematics
On the distribution of prime numbers in arithmetic progressions
{\displaystyle a} modulo q {\displaystyle q} . Dirichlet's theorem on primes in arithmetic progressions then tells us that π ( x ; q , a ) ∼ π ( x ) φ
Elliott–Halberstam_conjecture
Theorem in combinatorics on words
only if S differs by a finite set from a finite union of arithmetic progressions. The theorem was proved by Alan Cobham in 1969 and has since given rise
Cobham's_theorem
Several textbooks of number theory
Supplement VI. Primes in arithmetic progressions Supplement VII. Some theorems from the theory of circle division Supplement VIII. On the Pell equation Supplement
Vorlesungen über Zahlentheorie
Vorlesungen_über_Zahlentheorie
Concept in number theory
then Szemerédi's theorem states that S contains arbitrarily large finite arithmetic progressions, and the Furstenberg–Sárközy theorem states that some
Natural_density
grid so that no three of them lie on a line? Rudin's conjecture on the number of squares in finite arithmetic progressions The sunflower conjecture – can
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
of the reciprocals 1/ a(k) diverges. The Erdős conjecture on arithmetic progressions states that if the sum of the reciprocals of the members of a
List_of_sums_of_reciprocals
Function in discrete mathematics
the star denotes complex conjugation. The Plancherel theorem is a special case of Parseval's theorem and states: ∑ n = 0 N − 1 | x n | 2 = 1 N ∑ k = 0 N
Discrete_Fourier_transform
Potential counterexample to the generalized Riemann hypothesis
L(s,\chi )} has no zeros in this region. The prime number theorem for arithmetic progressions is equivalent (in a certain sense) to L ( 1 + i t , χ ) ≠
Siegel_zero
Japanese mathematician (born 1948)
Katsuda, Sunada also established a geometric analogue of Dirichlet's theorem on arithmetic progressions in the context of dynamical systems (1988). One can
Toshikazu_Sunada
Norwegian mathematician (1917–2007)
Selbert, Atle (April 1949). "An Elementary Proof of Dirichlet's Theorem About Primes in Arithmetic Progression". Annals of Mathematics. 50 (2): 297–304. doi:10
Atle_Selberg
the general public. Peter Gustav Lejeune Dirichlet publishes Dirichlet's theorem on arithmetic progressions, using mathematical analysis concepts to tackle
1837_in_science
French mathematician and chess player (1837–1916)
used by Adrien-Marie Legendre in his attempt to prove Dirichlet's theorem on arithmetic progressions. Lucas (1891, pp. 181–195) describes Moreau's analysis
Charles_Paul_Narcisse_Moreau
List of terms created from a person's name
Fermi–Dirac statistics Johann Dirichlet, German mathematician – Dirichlet function, Dirichlet's theorem on arithmetic progressions Walt Disney, American animator
List_of_eponyms_(A–K)
Area of a right triangle with rational-numbered sides
be shown (as an application of Dirichlet's theorem on primes in arithmetic progression) that the only torsion points on this elliptic curve are those with
Congruent_number
Number, approximately 3.14
complex numbers at which exp z is equal to one is then an (imaginary) arithmetic progression of the form: { … , − 2 π i , 0 , 2 π i , 4 π i , … } = { 2 π k i
Pi
German mathematician (1849–1917)
groups over any number field) generalizes Dirichlet's classical result about primes in arithmetic progressions. The study of Galois groups of infinite-degree
Ferdinand_Georg_Frobenius
Conjecture about prime numbers
H. Prime triplet Green–Tao theorem First Hardy–Littlewood conjecture Prime constellation Primes in arithmetic progression Ribenboim, Paulo (1996) [1988]
Dickson's_conjecture
Italian mathematician (born 1940)
applications of the large sieve method. It improves Dirichlet's theorem on prime numbers in arithmetic progressions, by showing that by averaging over the modulus
Enrico_Bombieri
Theorem for n = 14. 1835 — Lejeune Dirichlet proves Dirichlet's theorem about prime numbers in arithmetic progressions. 1859 — Bernhard Riemann formulates
Timeline_of_number_theory
Arithmetic progression – a sequence of numbers such that the difference between the consecutive terms is constant Generalized arithmetic progression –
List_of_real_analysis_topics
Integer that is a perfect square modulo some integer
regularities. Using Dirichlet's theorem on primes in arithmetic progressions, the law of quadratic reciprocity, and the Chinese remainder theorem (CRT) it is
Quadratic_residue
Finite extension of the rationals
encode the arithmetic behavior of Q {\displaystyle \mathbb {Q} } and K {\displaystyle K} , respectively. For example, Dirichlet's theorem asserts that
Algebraic_number_field
Press. On the statistical level, the classical result on primes in arithmetic progressions of Dirichlet generalises to Chebotaryov's density theorem; what
Non-abelian class field theory
Non-abelian_class_field_theory
Infinite sum
each the product of an element of an arithmetic progression with the corresponding element of a geometric progression. Example: 3 + 5 2 + 7 4 + 9 8 + 11
Series_(mathematics)
Jewish-Polish mathematician
the Siegel–Walfisz theorem, from which the prime number theorem for arithmetic progressions can be deduced. By using estimates on exponential sums due
Arnold_Walfisz
Mathematical sequence satisfying a specific pattern
multiplication of the elements of a geometric progression with the corresponding elements of an arithmetic progression. The nth element of an arithmetico-geometric
Arithmetico-geometric sequence
Arithmetico-geometric_sequence
harmonic progression In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression
Glossary_of_calculus
Infinite series that is not convergent
finite series such as arithmetic progressions in an infinite context. For instance, using this method, the sum of the progression 1 + 2 + 3 + … {\displaystyle
Divergent_series
Divergent sum of positive unit fractions
quadraturae arithmeticae, seu De additione fractionum [New arithmetic quadrature (i.e., integration), or On the addition of fractions] (in Latin). Bologna: Giacomo
Harmonic_series_(mathematics)
Particular kind of exponential sum
the Riemann zeta function, primes in short intervals, primes in arithmetic progressions, the spectral theory of automorphic functions and related topics
Kloosterman_sum
Rational number sequence
{5}{66}}n} Indeed [if] one will have examined diligently the law of arithmetic progression there, one will also be able to continue the same without these
Bernoulli_number
Russian mathematician (1937–2008)
SSSR, Ser. Mat. 54 (2): 303–315. Karatsuba, A. A. (1993). "On the zeros of arithmetic Dirichlet series without Euler product". Izv. Ross. Akad. Nauk, Ser
Anatoly_Karatsuba
Formal power series
1, another useful formula providing somewhat reversed floored arithmetic progressions — effectively repeating each coefficient m times — are generated
Generating_function
Infinite binary sequence generated by repeated complementation and concatenation
in arithmetic progression can be partitioned into two sets with equal sums. This follows directly from the expansion given by the binomial theorem applied
Thue–Morse_sequence
Numbers whose prime factors all divide the number more than once
are k-powerful numbers in an arithmetic progression. Moreover, if a1, a2, ..., as are k-powerful in an arithmetic progression with common difference d, then
Powerful_number
the proof is a generalization of Dirichlet's proof that there are infinitely many primes in arithmetic progressions.) But a prime represents a trivial
Class_formation
Infinite graph containing all countable graphs
{\displaystyle V} form a periodic sequence, so by Dirichlet's theorem on primes in arithmetic progressions this number-theoretic graph has the extension property
Rado_graph
Difference between logarithm and harmonic series
ISBN 0-201-89683-4. Lehmer, D. H. (1975). "Euler constants for arithmetical progressions" (PDF). Acta Arith. 27 (1): 125–142. doi:10.4064/aa-27-1-125-142
Euler's_constant
Polynomial sequence
Stirling polynomial Polynomials calculating sums of powers of arithmetic progressions Hurtado Benavides, Miguel Ángel. (2020). De las sumas de potencias
Bernoulli_polynomials
Operation on formal power series
1} , another useful formula providing somewhat reversed floored arithmetic progressions are generated by the identity ∑ n ≥ 0 f ⌊ n m ⌋ z n = 1 − z m 1
Generating function transformation
Generating_function_transformation
DIRICHLETS THEOREM-ON-ARITHMETIC-PROGRESSIONS
DIRICHLETS THEOREM-ON-ARITHMETIC-PROGRESSIONS
Male
French
French form of Latin Leo, LÉON means "lion."
Girl/Female
Australian, Greek
Watcher
Girl/Female
Hindu, Indian
Going on
Boy/Male
Hindu, Indian, Punjabi, Sikh
One on whom There is God's Grace
Biblical
an offering dedicated to God
Girl/Female
German, Greek, Swedish
Harvester
Girl/Female
Greek
God's name.
Boy/Male
Indian, Punjabi, Sikh
One on whom There is God's Grace
Girl/Female
Tamil
Udipti | உதிபà¯à®¤à¯€
On fire
Udipti | உதிபà¯à®¤à¯€
Girl/Female
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
On Fire
Boy/Male
Sikh
One on whom there is gods grace, Gods mercy
Girl/Female
Biblical
An offering dedicated to God.
Boy/Male
Sikh
One on whom there is gods grace, Gods mercy
Girl/Female
Australian, Danish, Greek, Netherlands
Name of God
Girl/Female
Arabic
Happines
Girl/Female
Greek
Watcher.
Surname or Lastname
English and Scandinavian
English and Scandinavian : variant of Thor.French (Thoré) : nickname for a strong or violent individual, from Old French t(h)or(el) ‘bull’. Compare Spanish Toro.French (Thoré) : from a reduced pet form of the personal name Maturin.
Boy/Male
Australian, Biblical, British, Christian, English
Pain; Force; Iniquity
Boy/Male
Indian, Punjabi, Sikh
One on whom There is God's Grace
Girl/Female
Egyptian
Great.
DIRICHLETS THEOREM-ON-ARITHMETIC-PROGRESSIONS
DIRICHLETS THEOREM-ON-ARITHMETIC-PROGRESSIONS
Boy/Male
Hebrew American Scottish
Gift from God.
Surname or Lastname
English
English : from a Middle English personal name, probably a pet form of Arnold, although Reaney has it as a survival of the Old English personal names Earngēat (male) ‘eagle Geat’ (a tribal name) or Earnḡ{dh} (female) ‘eagle battle’.Variant of French Arnette.
Boy/Male
Australian, French
Of Mars; The God of War
Girl/Female
Hindu
Contentment, Complete satisfaction
Surname or Lastname
English (Gloucestershire)
English (Gloucestershire) : variant spelling of Uzzell.
Boy/Male
Indian, Tamil
Handsome
Girl/Female
Indian
Futures of parent
Boy/Male
Hindu, Indian, Kannada, Marathi, Telugu
A Mythological King
Boy/Male
Gujarati, Hindu, Indian, Kannada, Sanskrit, Telugu
Ocean of Mercy
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Peace
DIRICHLETS THEOREM-ON-ARITHMETIC-PROGRESSIONS
DIRICHLETS THEOREM-ON-ARITHMETIC-PROGRESSIONS
DIRICHLETS THEOREM-ON-ARITHMETIC-PROGRESSIONS
DIRICHLETS THEOREM-ON-ARITHMETIC-PROGRESSIONS
DIRICHLETS THEOREM-ON-ARITHMETIC-PROGRESSIONS
prep.
In the service of; connected with; of the number of; as, he is on a newspaper; on a committee.
a.
Relating to, or skilled in, theory; theoretically skilled.
prep.
Denoting performance or action by contact with the surface, upper part, or outside of anything; hence, by means of; with; as, to play on a violin or piano. Hence, figuratively, to work on one's feelings; to make an impression on the mind.
prep.
To the account of; -- denoting imprecation or invocation, or coming to, falling, or resting upon; as, on us be all the blame; a curse on him.
prep.
In addition to; besides; -- indicating multiplication or succession in a series; as, heaps on heaps; mischief on mischief; loss on loss; thought on thought.
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
prep.
In progress; proceeding; as, a game is on.
prep.
Occupied with; in the performance of; as, only three officers are on duty; on a journey.
a.
Of or pertaining to arithmetic; according to the rules or method of arithmetic.
prep.
At or near; adjacent to; -- indicating situation, place, or position; as, on the one hand, on the other hand; the fleet is on the American coast.
pl.
of Theory
n.
Arithmetic.
prep.
Forward, in progression; onward; -- usually with a verb of motion; as, move on; go on.
prep.
In reference or relation to; as, on our part expect punctuality; a satire on society.
prep.
In continuance; without interruption or ceasing; as, sleep on, take your ease; say on; sing on.
n.
One skilled in arithmetic.
prep.
Indicating dependence or reliance; with confidence in; as, to depend on a person for assistance; to rely on; hence, indicating the ground or support of anything; as, he will promise on certain conditions; to bet on a horse.
prep.
At, or in contact with, the surface or upper part of a thing, and supported by it; placed or lying in contact with the surface; as, the book lies on the table, which stands on the floor of a house on an island.
n.
Speculation; theory.
v. t.
To formulate into a theorem.