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Class of integer sequences in mathematics
In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials
Elliptic divisibility sequence
Elliptic_divisibility_sequence
Type of integer sequence
coprime then this is a strong divisibility sequence. Elliptic divisibility sequences are another class of divisibility sequences. Everest, Graham; van der
Divisibility_sequence
On prime divisors of differences two nth powers
Lehmer sequences are examples of divisibility sequences. It is also known that if ( W n ) n ≥ 1 {\displaystyle (W_{n})_{n\geq 1}} is an elliptic divisibility
Zsigmondy's_theorem
Numbers obtained by adding the two previous ones
Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property gcd ( F
Fibonacci_sequence
Topics referred to by the same term
strana), a Czech political party Electrodynamic suspension Elliptic divisibility sequence Energy-dispersive X-ray spectroscopy Effluent decontamination
EDS
2y\mathbb {Z} [x,A,B]} . The division polynomials form a generic elliptic divisibility sequence over the ring Q [ x , y , A , B ] / ( y 2 − x 3 − A x − B )
Division_polynomials
theorem Congruent number Arithmetic of abelian varieties Elliptic divisibility sequences Mordell curve Fermat's Last Theorem Mordell conjecture Euler's sum
List_of_number_theory_topics
Methods to test or prove primality
In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods
Elliptic_curve_primality
Integer divisible by sum of its digits
that bn − 1 is divisible by all digit sums in the sequence, then the divisibility by those sums is maintained. If our initial sequence is chosen so that
Harshad_number
Number divisible only by 1 and itself
Eisenstein's criterion, a test for whether a polynomial is irreducible based on divisibility of its coefficients by a prime number and its square. The concept of
Prime_number
Decomposition of a number into a product
computer science have been brought to bear on this problem, including elliptic curves, algebraic number theory, and quantum computing. Not all numbers
Integer_factorization
Recursive integer sequence
Press, ISBN 978-0-19-533454-8 Koshy, Thomas & Zhenguang Gao (2011) "Some divisibility properties of Catalan numbers", Mathematical Gazette 95:96–102. Larcombe
Catalan_number
Integer having a non-trivial divisor
15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36. (sequence A002808 in the OEIS) Every composite number can be written as the product
Composite_number
Prime number of the form 2^n – 1
77232917, 82589933, 136279841. (sequence A000043 in the OEIS) Since they are prime numbers, Mersenne primes are divisible only by 1 and themselves. However
Mersenne_prime
Class of natural numbers with many divisors
composite numbers have often been used as radices, due to their high divisibility for their size. For example: Binary (base 2) Senary (base 6) Duodecimal
Superior highly composite number
Superior_highly_composite_number
Numbers with many divisors
The first 41 highly composite numbers are listed in the table below (sequence A002182 in the OEIS). The number of divisors is given in the column labeled
Highly_composite_number
Number of the form (n * 2^n) - 1
= 7, and W512 = M521. Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides W(p + 1) / 2
Woodall_number
Sum of a number's digits
used for quick divisibility tests: a natural number is divisible by 3 or 9 if and only if its digit sum (or digital root) is divisible by 3 or 9, respectively
Digit_sum
Natural number
(ed.). "Sequence A000129 (Pell numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10. "Jacobi elliptic functions
12_(number)
Numbers that contain only the digit 1
any m and n. That is, the repunits of a fixed base form a strong divisibility sequence. As a consequence, If m and n are relatively prime, Rm(b) and Rn(b)
Repunit
Prime such that p^2 divides 2^(p-1)-1
congruence and divisibility of h+" (PDF), Acta Arithmetica, 71 (1): 55–64, doi:10.4064/aa-71-1-55-64 Jakubec, S. (1998), "On divisibility of the class number
Wieferich_prime
Number that remains the same when its digits are reversed
131, 151, ... (sequence A002385 in the OEIS). The palindromic square numbers are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, ... (sequence A002779 in the
Palindromic_number
Mathematical result in differential geometry
proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related
Atiyah–Singer_index_theorem
Integer whose representation contains every digit in its number base
have redundant digits. The sum of the digits 0 to 9 is 45, passing the divisibility rule for both 3 and 9. The first base 10 pandigital prime is 10123457689;
Pandigital_number
Result of multiplying six instances of a number
curves, which are elliptic curves of the form y 2 = x 3 + k . {\displaystyle y^{2}=x^{3}+k.} When k {\displaystyle k} is divisible by a sixth power, this
Sixth_power
Number used for counting
and b. This Euclidean division is key to the several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory
Natural_number
Branch of pure mathematics
(Divisibility Tests), p. 102–108 Ore, Oystein (1948). Number Theory and Its History (1st ed.). McGraw-Hill. Watkins, John J. (2014). "Divisibility".
Number_theory
Natural number
strictly non-palindromic number. It is the smallest conductor of a rank 2 Elliptic curve. 390 = 2 × 3 × 5 × 13. It is a nontotient and the sum of four consecutive
300_(number)
Pair of integers related by their divisors
original on 2022-09-25. Retrieved 2020-09-07. Lee, Elvin (1969). "On Divisibility by Nine of the Sums of Even Amicable Pairs". Mathematics of Computation
Amicable_numbers
Infinite integer series where the next number is the sum of the two preceding it
Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the
Lucas_number
Group of real 2×2 matrices with unit determinant
n Z < Z ≅ π1 (PSL(2, R)), which form a lattice of covering groups by divisibility; these cover SL(2, R) if and only if n is even. The center of SL(2, R)
SL2(R)
Positive integer of the form (2^(2^n))+1
4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ... (sequence A000215 in the OEIS). If 2k + 1 is prime and k > 0, then k itself must
Fermat_number
Integer having only small prime factors
handwritten note. Naccache, David; Shparlinski, Igor (17 October 2008). "Divisibility, Smoothness and Cryptographic Applications" (PDF). eprint.iacr.org. arXiv:0810
Smooth_number
Abundant number whose proper divisors are all deficient numbers
abundant numbers are: 20, 70, 88, 104, 272, 304, 368, 464, 550, 572 ... (sequence A071395 in the OEIS) The smallest odd primitive abundant number is 945
Primitive_abundant_number
Number whose divisors summed twice over equal twice itself
superperfect numbers are: 2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... (sequence A019279 in the OEIS). To illustrate: it can be seen that 16 is a superperfect
Superperfect_number
Infinite sum
its sequence of partial sums. Either the sequence of partial sums or the sequence of terms completely characterizes the series, and the sequence of terms
Series_(mathematics)
Repeated sum of a number's digits
remainder upon division by 9 will be 0), which allows it to be used as a divisibility rule. The formula for the function d r b : N → ⋃ k = 0 b − 1 { k }
Digital_root
Number equal to the sum of its proper divisors
function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence. All perfect numbers are also S {\displaystyle
Perfect_number
Decentralized digital cryptocurrency
smart contracts and Lightning Network. Before, bitcoin only used a custom elliptic curve with the ECDSA algorithm to produce signatures. In September 2021
Bitcoin
Number sequence 3,0,2,3,2,5,5,7,10,...
P(n) divisible by composite index n was found only in 1982 by William Adams and Daniel Shanks. They presented a detailed investigation of the sequence, with
Perrin_number
Number that is less than the sum of its proper divisors
divisible by the first k primes (sequence A047802 in the OEIS). If A ( k ) {\displaystyle A(k)} represents the smallest abundant number not divisible
Abundant_number
Number of form 2^(2^p-1)-1 with prime exponent
(E. C.) L. E. Dickson, History of the theory of numbers. Volume 1: Divisibility and primality (1919). Published by Washington, Carnegie Institution of
Double_Mersenne_number
Product of two prime numbers
51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, and 95 (sequence A001358 in the OEIS) Semiprimes that are not square numbers are called
Semiprime
Positive integer with large prime factors
Number". MathWorld. Finch's definition from Number Theory Archives "Divisibility, Smoothness and Cryptographic Applications", D. Naccache and I. E. Shparlinski
Rough_number
Integer side lengths of a right triangle
to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics, vol. 97, Springer, p. 3, ISBN 9780387979663. Sloane, N. J. A. (ed.), "Sequence A237518
Pythagorean_triple
Number equal to the sum of all or some of its divisors
36, 40, ... (sequence A005835 in the OEIS) Every multiple of a semiperfect number is semiperfect. A semiperfect number not divisible by any smaller
Semiperfect_number
Prime number one less or more than a factorial
first 10 factorial primes (for n = 1, 2, 3, 4, 6, 7, 11, 12, 14) are (sequence A088054 in the OEIS): 2 (0! + 1 or 1! + 1), 3 (2! + 1), 5 (3! − 1), 7 (3
Factorial_prime
Mathematical concept
6679881 (sequence A005849 in the OEIS). Still, it is conjectured that there are infinitely many Cullen primes. A Cullen number Cn is divisible by p = 2n − 1
Cullen_number
Commutative ring with no zero divisors other than zero
the ring of integers and provide a setting that is useful for studying divisibility. "Integral domain" is defined almost universally as above, but there
Integral_domain
Conjecture on zeros of the zeta function
of the simplicial complex determined by the lattice of integers under divisibility is o ( n 1 / 2 + ϵ ) {\displaystyle o(n^{1/2+\epsilon })} for all ϵ >
Riemann_hypothesis
Numbers with special prime factorization
3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000 (sequence A052486 in the OEIS). The sequence grows as O(n2/log log n), and the sum of reciprocals
Achilles_number
Integer of the form 3 × 2^n – 1 for non-negative n
6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ... (sequence A055010 in the OEIS) The 9th century mathematician, physician, astronomer
Thabit_number
Two raised to an integer power
non-negative values of n are: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... (sequence A000079 in the OEIS) By comparison, powers of two with negative exponents
Power_of_two
Number that has a perfect number of factors adding up to another perfect number
numbers: 12 and (2126)(261 − 1)(231 − 1)(219 − 1)(27 − 1)(25 − 1)(23 − 1) (sequence A081357 in the OEIS). The second of these has 76 decimal digits: 6,086
Sublime_number
Theory in abstract algebra
often used in the context of elliptic curves. Let E / K {\displaystyle E/K} be an elliptic curve. There is a short exact sequence 0 → E [ m ] → E → P ↦ m ⋅
Kummer_theory
Positive integer whose divisors have a harmonic mean that is an integer
numbers are 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS). Harmonic divisor numbers were introduced by Øystein
Harmonic_divisor_number
Number that is more than the sum of its proper divisors
Dickson, Leonard Eugene (1919). History of the Theory of Numbers, Vol. I: Divisibility and Primality. Carnegie Institute of Washington. Prielipp, Robert W.
Deficient_number
Numbers whose aliquot sums form a cyclic sequence
the proportion of the sums of the sociable number cycles divisible by 10 approaches 1 (sequence A292217 in the OEIS). P. Poulet, #4865, L'Intermédiaire
Sociable_number
Product of an integer with itself
alternative way in factorization of large numbers. Instead of testing for divisibility, test for squarity: for given m and some number k, if k2 − m is the square
Square_number
Number that has fewer digits than the number of digits in its prime factorization
base 10, 4 = 22, 6 = 2×3, 8 = 23, and 9 = 32 are extravagant numbers (sequence A046760 in the OEIS). There are infinitely many extravagant numbers in
Extravagant_number
Positive integer that is the product of three distinct prime numbers
every fourth consecutive positive integer is divisible by 4 = 2 × 2 and therefore not squarefree (sequence A291466 in the OEIS). The numbers 2013 = 3
Sphenic_number
Positive integer of the form 4n + 1
not divisible by a smaller Hilbert number (other than 1). The sequence of Hilbert primes begins 5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ... (sequence A057948
Hilbert_number
Positive integer that is an integer power of another positive integer
considered perfect powers (0k = 0 for any k > 0, 1k = 1 for any k). A sequence of perfect powers can be generated by iterating through the possible values
Perfect_power
Numbers that evenly divide powers of 60
45, 48, 50, 54, 60, ... (sequence A051037 in the OEIS) Several other sequences at the On-Line Encyclopedia of Integer Sequences have definitions involving
Regular_number
Topic in algebraic number theory
equation: a notable example is Siegel's theorem on integral points on elliptic curves, and more generally superelliptic curves of the form yn = f(x).
S-unit
Numbers whose prime factors all divide the number more than once
648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, ... (sequence A001694 in the OEIS). If m = a2b3, then every prime in the prime factorization
Powerful_number
Numeral ambigram
6969, 8008, 8118, 8698, 8888, 8968, 9006, 9116, 9696, 9886, 9966, … (sequence A000787 in the OEIS) The first few strobogrammatic primes are: 11, 101
Strobogrammatic_number
Number-theoretical function
"Introduction". Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions. Springer Science &
Sum_of_squares_function
Number whose sums of distinct divisors represent all smaller numbers
2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2. The sequence of practical numbers (sequence A005153 in the OEIS) begins 1, 2, 4, 6, 8, 12, 16, 18,
Practical_number
Number whose divisors add to a multiple of that number
45532800, 142990848, 459818240,14182439040, 4364925090, 8589869056 ... (sequence A007691 in the OEIS). The sum of the divisors of 120 is 1 + 2 + 3 + 4 +
Multiply_perfect_number
Numbers whose sum of divisors is twice the number minus 1
known almost perfect numbers are powers of 2 with non-negative exponents (sequence A000079 in the OEIS). Therefore the only known odd almost perfect number
Almost_perfect_number
Magma obeying the Latin square property
the geometric notion of extended Steiner triple, also called Generalized Elliptic Cubic Curve (GECC). A quasigroup (Q, ∗) is called weakly totally anti-symmetric
Quasigroup
Natural number whose divisor sum is greater than that of any smaller number
are 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... (sequence A002093 in the OEIS). For instance, 5 is not highly abundant because σ(5)
Highly_abundant_number
Integer named after Reo Fortune
37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, ... (sequence A005235 in the OEIS). The Fortunate numbers sorted in numerical order with
Fortunate_number
Number whose first n digits is a multiple of n
{\displaystyle bk} and b ( k + 1 ) − 1 {\displaystyle b(k+1)-1} that is divisible by n {\displaystyle n} . If n {\displaystyle n} is less or equal to b
Polydivisible_number
Type of natural number
5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A004490 in the OEIS) are also the first 15 superior highly composite numbers
Colossally_abundant_number
Integer where the average of its positive divisors is also an integer
and 2, and their average 3/2 is not an integer. The first numbers in the sequence of arithmetic numbers are 1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21
Arithmetic_number
Type of composite integer
706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985. (sequence A006753 in the OEIS) W.L. McDaniel in 1987 proved that there are infinitely
Smith_number
Number with a half-integer abundancy index
4320, 4680, 26208, 8910720, 17428320, 20427264, 91963648, 197064960, ... (sequence A159907 in the OEIS) 24 is a hemiperfect number because the sum of the
Hemiperfect_number
some constant C ( ε ) {\displaystyle C(\varepsilon )} such that, for any elliptic curve E {\displaystyle E} defined over Q {\displaystyle \mathbb {Q} } with
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Natural number with a decimal representation made of repeated instances of the same digit
infinitely many non-Brazilian primes, forming the sequence 2, 3, 5, 11, 17, 19, 23, 29, 37, 41, 47, 53, ... (sequence A220627 in the OEIS) If a Fermat number F
Repdigit
Product of prime numbers, plus one
2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, ... (sequence A006862 in the OEIS). The first few Kummer numbers are 1, 5, 29, 209, 2309
Euclid_number
Integer sequence in number theory
would be an odd perfect number. If n is a cube-free Descartes number not divisible by 3, then n has over one million distinct prime divisors. If D = p q
Descartes_number
Type of positive composite integer
\log X\right)^{2}}} . Thomas Wright (2018). "There are infinitely many elliptic Carmichael numbers". Bull. London Math. Soc. 50 (5): 791–800. arXiv:1609
Lucas–Carmichael_number
Rational right triangles cannot have square area
(called a congruum) cannot itself be square. The only rational points on the elliptic curve y 2 = x ( x − 1 ) ( x + 1 ) {\displaystyle y^{2}=x(x-1)(x+1)} are
Fermat's right triangle theorem
Fermat's_right_triangle_theorem
Computation modulo a fixed integer
defined by the divisibility by m and because −1 is a unit in the ring of integers, a number is divisible by −m exactly if it is divisible by m. This means
Modular_arithmetic
Integer whose multiples are digit rotations
necessary structure given in the next section. Allowing leading zeros, the sequence of cyclic numbers begins: (106 − 1) / 7 = 142857 (6 digits) (1016 − 1)
Cyclic_number
Number n where the highest prime factor of (n^2 + 1) is at least 2n
85, 86, 87, 88, 89, 90, 92, 94, 95, 96, 97... (sequence A005528 in the OEIS). The complementary sequence (numbers below 100 that aren't Størmer) is only
Størmer_number
Type of positive integer
following property: there exists a positive integer a such that in the sequence (a, a + 1, …, a + k) of consecutive integers, each of the elements has
Erdős–Woods_number
Power of a prime number
233, 239, 241, 243, 251, ... (sequence A246655 in the OEIS). The prime powers are those positive integers that are divisible by exactly one prime number;
Prime_power
Class of numbers not containing a particular letter in English
through 319, ... (sequence A008554 in the OEIS). For 1<N<109, aban numbers are numbers which the integer part of N/1000 is divisible by 1000. Eban numbers
Ban_number
Integer divisible by the number of its divisors
A refactorable number or tau number is an integer n that is divisible by the count of its divisors, or to put it algebraically, n is such that τ ( n )
Refactorable_number
Sheaf cohomology on the étale site
with constant coefficients of order divisible by the characteristic in a similar way, using the Artin–Schreier sequence 0 → Z / p Z → K → x ↦ x p − x
Étale_cohomology
Probable prime that is composite
that are coprime to x is called a Carmichael number. Catalan pseudoprime Elliptic pseudoprime Euler pseudoprime Euler–Jacobi pseudoprime Fermat pseudoprime
Pseudoprime
Same digit count as prime factorization
example, in base 10, 1, 2, 3, 5, 7, and 10 (2 × 5) are equidigital numbers (sequence A046758 in the OEIS). All prime numbers are equidigital numbers in any
Equidigital_number
Type of positive integer pairs
2295), (5775, 6128), which (removing brackets) may be sorted by magnitude (sequence A005276 in the OEIS). All known pairs of betrothed numbers have opposite
Betrothed_numbers
Number that cannot be written as an aliquot sum
324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ... (sequence A005114 in the OEIS). Unsolved problem in mathematics Are there any odd
Untouchable_number
Figurate number
The triangular numbers or triangle numbers are the sequence of positive integers that can be represented as a lattice of points arranged in an equilateral
Triangular_number
Type of natural number
values of k, together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of k-hyperperfect numbers: It can
Hyperperfect_number
Prime number of the form 2^u × 3^v + 1
209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, ... (sequence A005109 in the OEIS) It has been conjectured that there are infinitely
Pierpont_prime
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ELLIPTIC DIVISIBILITY-SEQUENCE
Boy/Male
Indian, Sanskrit
Order; Sequence
Surname or Lastname
English
English : variant of Douthwaite, a habitational name from Dowthwaite in Cumbria or Dowthwaite Hall in North Yorkshire. The first is from the Old Norse personal name Dúfa + Old Norse þveit ‘clearing’; the second is from the Old Irish personal name Dubhan + Old Norse þveit. The elliptic form of the surname probably reflects the local pronunciation of the place names.
Boy/Male
Indian, Sikh
Music; In-sequence
Boy/Male
Irish
From dubh “â€blackâ€â€ and lan “â€blade, swordâ€â€ means “â€black sword.â€â€ Dubhlainn loved the fairy queen and legendary harpist Aoibhell who gave him her cloak of invisibility to wear in battle.
Boy/Male
Irish
From dubh “â€blackâ€â€ and lan “â€blade, swordâ€â€ means “â€black sword.â€â€ Dubhlainn loved the fairy queen and legendary harpist Aoibhell who gave him her cloak of invisibility to wear in battle.
Surname or Lastname
English
English : patronymic for the son of a vicar or, perhaps in most cases, an occupational name for the servant of a vicar (see Vicker). In many cases it may represent an elliptical form of a topographic name. Compare Parsons.
Girl/Female
Tamil
Anuloma | அநà¯à®²à¯‹à®®à®¾
Sequence
Anuloma | அநà¯à®²à¯‹à®®à®¾
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Sequence
Surname or Lastname
English
English : from a medieval male personal name (from Latin Hilarius, a derivative of hilaris ‘cheerful’, ‘glad’, from Greek hilaros ‘propitious’, ‘joyful’). The Latin name was chosen by many early Christians to express their joy and hope of salvation, and was borne by several saints, including a 4th-century bishop of Poitiers noted for his vigorous resistance to the Arian heresy, and a 5th-century bishop of Arles. Largely due to veneration of the first of these, the name became popular in France in the forms Hilari and Hilaire, and was brought to England by the Norman conquerors.English : from the much rarer female personal name Eulalie (from Latin Eulalia, from Greek eulalos ‘eloquent’, literally well-speaking, chosen by early Christians as a reference to the gift of tongues), likewise introduced into England by the Normans. A St. Eulalia was crucified at Barcelona in the reign of the Emperor Diocletian and became the patron of that city. In England the name underwent dissimilation of the sequence -l-l- to -l-r- and the unfamiliar initial vowel was also mutilated, so that eventually the name was considered as no more than a feminine form of Hilary (of which the initial aspirate was in any case variable).
ELLIPTIC DIVISIBILITY-SEQUENCE
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ELLIPTIC DIVISIBILITY-SEQUENCE
ELLIPTIC DIVISIBILITY-SEQUENCE
ELLIPTIC DIVISIBILITY-SEQUENCE
ELLIPTIC DIVISIBILITY-SEQUENCE
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