Letter P

P_________________________________________________________________________________________________
Pappus' theorem
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parabola
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parabolic
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parabolic point of a surface
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parabolic spiral
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parabolic umbilic catastrophe
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PLcatastrophe.
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paraboloid
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paracompact
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paradox of counting
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We teach smallchildren to count the fingers on one hand as follows: "1, 2, 3, 4, 5". But these are cardinalnumber names (PL and see how cardinality is modeled byone type of brown-bag procedure, while ordinality is modeled by a very differentprocedure). For at least two critical reasons, we should teach the childto count fingers as follows: "first, second, third, fourth, fifth: five".These ordinal number names (PL) ordinate the fingersin the counting process and measure the cardinality, which, inthis case, is "five". The two critical reasons for this latter procedure isthat it avoids the paradox of counting (soon revealed) and connectswith many useful concepts and procedures (revealed below). Converselytraditional counting invokes this paradox and cuts the child offfrom these useful concepts and procedures. The paradox can beinvoked via a corny joke of our ancestors, in parody of a traditional syllogism:"The apostles were twelve; Peter was an apostle; therefore, Peter was twelve".Obviously, the collective name "twelve apostles" was transferred to aparticular "apostle". And we can translate the traditional countingprocedure as follows: "The fingers of a hand are five; little finger isa finger of the hand; therefore, little finger is five". This cuts the childoff from counting as inherent in any extensive measurement process.We lay off a rule, say, five times on an segment of the floor, and say,"the length of this segment is five feet". This procedure measures theinterval. Similarly, ordination measures cardination. We alsolose the connection with the choice or combinatoric formula for number ofdifferent ways of counting. We can choose the "first" in five (5) differentways; that leaves four (4) ways of choosing the "second"; three (3) ways ofchoosing the "third"; two (3) ways of choosing the "fourth"; leaving one wayof choosing the "fifth". Since these procedure function independently,we can multiply them to obtain a total: 5 * 4 * 3 * 2 * 1 = 120 ways ofcounting the fingers of a hand. (PL factorial.) Evenmore critically, the child is cut off from future understanding of Galileo'sparadox (PL). The consequence is that, if the individual studies"calculus" in high school or college, he or she is unprepared for the fact that-- unlike finite sets -- infinite sets can yield different resultswhen counted differently. Children can easily be taught the ordinal-cardinaldistinction by brownbag models andsee that an ordinal model taken apart presents the associated cardinal model.Having learned this, we can allow the (convenient) "abuse of language" (in the languageof logician, xxx yyy) of counting with ordinal names.
parallel
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parallelepiped
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parallelogram
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parallelogram identity
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parameter
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parameterized curves
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parametric equation
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parametric programming
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parity
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partial derivative
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partial differential equation
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partial fractions
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partially ordered set (poset)
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A set witha partial ordering (PL).
partial inverse operation
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An inverseoperation (PL) of arithmetic is partial on a given arithmetical numbersystem (PL) iff some result of the operation does not exist in the system(not "all-in-the-family"), equivalently, the system is not closed under thegiven operation. Thus, subtraction, division, logarithm, root extraction arepartial on the naturals; division, logarithm, root extraction on the integers;logarithm and root extraction on the rationals; and root extraction on thereals. Rendering total (non-partial) the inverse operationsof arithmetic results in extending the number systems of arithmetic beyond thenatural numbers.
partial order(ing)
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Since any orderingrelation (PL) must be transitive (PL), a partial orderingis an ordering O that is also
reflexive: a O a, for any a,
and antisymmetric: a O b and bO a iff a = b.
The best known formal partial orderingsare "less than or equal to" (<=) and "greater than or equal to" (>=). A partially orderedset is also known as a "poset". (PL total ordering.)
partorial
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#(n)is the nth partorial iff it is the product of the first n primorials (definedbelow), where #(0)=1, by definition. Thus, #(1) = 2,#(2) = 12, &(3) = 360, etc. (The name "partorial" derives from the result thatthe enumeration -- PL -- of #(n) provides acanonical partition -- PL -- of factorial !(n + 1).A partorial enumeration table is equivalent to a table of MacMahon's tableof inversions on the symmetric group (PL). Since the latter is known toprovide an alternative to the Cayley table (PL) for a finite group,it follows that this is so for a partorial table. The generator (PL) of thepartorial is also the generator of the Gaussian q-nomial (PL).
primorial
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@(n)is the nth primorial iff it is the product of the first n prime numbers, where@(0)=1, by definition. Thus, @(1) = 2, @(2) = 2 * 3 =6, &(3) = 2 * 3 * 5 = 30, etc. (PL partorial, above.)
partial sum
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particular solution
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partition of the factorial
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PL partorial.
partition of a positive integer
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partition of a set
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partition of unity
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Pascal's theorem
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path
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pathological
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payoff matrix
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Peano axioms
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Pecking Order lattice
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The complementeddistributive lattice (CPL) or distributive lattice (DL) -- PL both -- canbe modeled for Pre-School by the Mathtivity, "Pecking Order". Chickens in a barnyard form adominating hierarchy wherein a chicken may peck another chicken but not be pecked back asa sign of dominance. This can be graphed as a Hasse diagram (PL) for a lattice.The CPL is modeled as a simple Pecking Order(SPO); the DL, as a mixed Pecking Order. Later, inPrimary School, by changing labels, a SPO becomes a (CPL) factor lattice (PL),on a "square-free" number, say, 30; a MPO becomes a (DL) factor lattice on a "nonsquare-free",say, 12. Then, by changing labels, this expands into the Pythagorean Repertory (PL).
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Pecking Order repertory
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PLPythagorean repertory.
pedal coordinates
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pedal curve
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pedal equation
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pedal triangle
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Pell equation
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pencil
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pentagon
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A polygon (PL) withfive edges (PL).
percent
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The symbol "%" (labeled "per centum")denotes the implicit denominator of 100 in a parcentage. For example, 10% =10/100.
percentage
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A rational number equal to afraction with denominator one hundred. (PL percent.)
per centum
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PL percent.
percolation network
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perfect cube
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perfect field
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perfect graph
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perfect matching
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perfect number
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perfect set
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perfect square
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perimeter
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periodic
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periodic function
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periodic perturbation
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periphery
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permutation
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permutation group
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permutation matrix
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permutation symbol
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permutation tensor
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perpendicular
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Perron-Frobenius theorem
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Perron-Frobenius theory
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perspectivity
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perturbation
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perturbation theory
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Petersen graph
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Petersen's theorem
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Pfaffian differential equation
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p-group
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phase
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phase plane
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phase space
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pi
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Picard's big theorem
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Picard's little theorem
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piecewise
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pigeonhole principle
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piriform
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pivot
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pivoting
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place
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placeholder
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planar
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planar graph
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plane
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plane geometry
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plane graph
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plane group
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plane of reflection
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plane of symmetry
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plane trigonometry
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plateau problem
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Platonic graph
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Platonic solid
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Platonism
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plus
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Poincaré conjecture
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Poincaré recurrence theorem
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Poincaré's theorem
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Poinsot's spiral
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point
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point of inflection
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point set topology
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pointwise convergence
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Poisson's equation
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polar
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polar axis
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polar coordinates
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polarity
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polar vector
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pole
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Pollard p-1 factorization
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A prime factorization algorithm which is implementable in one- ortwo-step form. Single-step: if p - 1 is a product of small primes, discover a number m c^q (mod n) s.t. a large number q divides p - 1 without remainder (p-1|q) and c, n; (c, n)=1 (meaning coprime: with only unity as common factor). Then m 1 (mod n), so p|m - 1. Now, on the strongpossibility that n|m - 1 (doesn't evenly divide), then GCD(m - 1, n) (greatest commondivisor) is a nontrivial divisor of n. Double-step: prime canbe factored if p - 1 is a product of small primes and a single largeprime.
polyalgorithm
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polygon
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A 2-D geometricstructure consisting of n vertices (PL) such that any twois connected by a line segment or edge (PL) whereby its plane is separated into an "inside"of the structure and an "outside".
polygonal
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Adjectival description ofa polygon (PL).
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polyhedral
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Adjectival description ofa polyhedron (PL).
polyhedral angle
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The angle at the vertexof a polyhedron (PL).
polyhedron
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A 3-D geometricstructure consisting of n vertices (PL) such that any twois connected by a line segment or edge (PL) whereby its space is separated into an "inside"of the structure and an "outside".
polynomial
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pons asinorum
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poset
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PL partial order.
positional notation
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position vector
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positive
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positive angle
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positive axis
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positive definite
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positive matrix
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positive real function
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positive set
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potential
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potential theory
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power
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power axiom
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power set
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The set composed of allsubsets of a given set, including the empty set (PL) and the set itself.
precision
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precompact set
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predecessor
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The ordon (PL) precedinga given ordon in an ordering.
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predicate
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predicate calculus
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pre-Hilbert space
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preimage
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primary
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PL standard dictinary.
prime ceiling
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The number of primesnot exceeding number x, denoted p(x).
prime counting function
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p(n) for number of primes n. Thus, for 2, 3, 5, 7, p(n) = 4. G. H. Hardy (`877-1947) & E. M. Wright estimate (using floor function PL): p(n) = - 1 S_j=3ⁿ[(j - 2)! - j |_ (j - 2)!/j _|]. The asymptotic value, PL:p(n) _~ li(x), where li(x) is the logarithmic integral, PL. Also, PL asymptotic prime numbertheorem.
prime factorization algorithms
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Many,diverse in complexity. The "direct search method", by trial divisors is simplest, butpractical only for small numbers. The fastest known deterministic algorithm is thePollard-Strassen. PL Pollard p-1 factorization, Dixon'sfactorization method, continued fraction factorization algorithm.
prime floor
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The number of primesnot less than the square root of a number , denoted p(x).

prime number
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A natural number orinteger is primal iff it has no proper factors, that is, its only factorsare 1 and the number itself.

prime number ("bonus") algorithm (Hays)

Given the natural number sequence, N = 1, 2, 3, ..., n, ...:
extract the subset of the first k primes, P = {p₁, p₂, ..., p_k};
and form the set O mod p_i, i = 1, 2, ..., k, whose simple ordering has the sequence, O' = q₁, q₂, ..., in which no q_j is divisible by any p_i;
and form the simple ordering of P O.
Forming the simple ordering, M, of P O, then, in its sequence, M' = p₁, p₂, ..., p_k, q₁, ..., we find the new prime p_k+1 = q₁. In other words, application (to N) of each sieving step yields the next prime.
Proof: This follows from a special case of Euler's totient function, ( ), involving coprimality, that is, sharing a common prime factor.
TOTIENT: Given prime number p, the numerosity of numbers less than and coprime to p is given as (p) = p(1 - 1/p) = p - 1.
However, in k applications of sieving to N, these p_k - 1 numbers are all sieved out between p_k and the first noncoprime of any p_i, i = 1, 2, ..., k, namely, q₁ , which points to q₁ (in M') as the first number in the sequence not divisible by any p_i, i = 1, 2, ..., k.
In other words, in M', q₁ = p_k+1.
Historically, the Erastothenes sieve has been known to find prime numbers by elminating every multiple of any prime found. Implicitly, this means that, by finding a prime and eliminating its mulltiples, sieving also finds the next prime. The above algorithm makes this "bonus" explicit. Hence, the Sieving Algorithm becomes a Prime Number Algorithm. (If improvements can be made on recursion of the modular operator -- a kind of extension of "The Chinese Remainder Theorem" -- then finding primes can be considerably simplified.)
PL asymptotic prime number theorem.
prime stretch
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The difference betweenthe prime ceiling and the prime floor: p(x) - p(x). Useful in writing sieving(PL) asa clusion (PL) in terms of prime factors, p_i, i = 1,..., r: p(N) - p(N) + 1 = N - [(N/p₁ + (N/p₂ + ... + (N/p_r] +[(N/p₁p₂) + (N/p₁p₃) + ... + (N/p_r-1p_r] - [(N/p₁p₂p₃) + ...] + ... + (-1)^r-1(N/(p₁p₂...p_r))..
primitive
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primitive circle
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primitive element
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primitive period
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primitive polynomial
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primitive Pythagorean triple
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primitive root of unity
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principal branch
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principal curvatures
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principal ideal
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principal normal
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principal value
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principle of the maximum
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principle of the minimum
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prism
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prismatic surface
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prismatoid
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prismoid
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probability space
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probability theory
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process
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product
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product measure
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progression
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projection
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projective geometry
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projective group
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projective line
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
projective module
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
projective plane
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
projective space
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
projectivity
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
prolate
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
prolate cycloid
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
proof
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
proper
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
proper character
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
proper class
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
proper fraction
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
proportion
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
proportional
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
proportional parts
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
proposition
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
propositional calculus
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
pseudorandom numbers
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
pseudo-Riemannian manifold
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
psi function
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
public key cryptography
.
Wherein the encoding key is disclosed w/o compromising the encoded message. The two best-knownmethods are the knapsack problem (PL) and RSA encryption (PL). PL also Flanneryalgorithm (a.k.a. Cayley-Purser a.).
punctured neighborhood
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
pure geometry
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
pure imaginary number
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
pure mathematics
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
pure projective geometry
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
pyramid
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
pyramidal surface
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
Pythagorean repertory
.
PL PECKING-ORDERrepertory.
Pythagorean theorem
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.
Pythagorean triple
.
May beread at http://www.harcourt.com/dictionary /browse/19/
.