The Experts below are selected from a list of 69 Experts worldwide ranked by ideXlab platform
Pallab Kanti Dey - One of the best experts on this subject based on the ideXlab platform.
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prime powers dividing products of Consecutive Integer values of x 2 n 1 x2n 1
Research in Number Theory, 2020Co-Authors: Stephan Baier, Pallab Kanti DeyAbstract:Let n be a positive Integer and $$f(x) := x^{2^n}+1$$. In this paper, we study orders of primes dividing products of the form $$P_{m,n}:=f(1)f(2)\ldots f(m)$$. We prove that if $$m > \max \{10^{12},4^{n+1}\}$$, then there exists a prime divisor p of $$P_{m,n}$$ such that $$\mathrm{ord}_{p}(P_{m,n} )\le n\cdot 2^{n-1}$$. For $$n=2$$, we establish that for every positive Integer m, there exists a prime divisor p of $$P_{m,2}$$ such that $$\mathrm{ord}_{p} (P_{m,2}) \le 4$$. Consequently, $$P_{m,2}$$ is never a fifth or higher power. This extends work of Cilleruelo [6] who studied the case $$n=1$$.
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prime powers dividing products of Consecutive Integer values of x 2 n 1
arXiv: Number Theory, 2019Co-Authors: Stephan Baier, Pallab Kanti DeyAbstract:Let $n$ be a positive Integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)\cdots f(m)$. We prove that if $m > \max\{10^{12},4^{n+1}\}$, then there exists a prime divisor $p$ of $P_{m,n}$ such that ord$_{p}(P_{m,n} )\leq n\cdot 2^{n-1}$. For $n=2$, we establish that for every positive Integer $m$, there exists a prime divisor $p$ of $P_{m,2}$ such that ord$_{p} (P_{m,2}) \leq 4$. Consequently, $P_{m,2}$ is never a fifth or higher power. This extends work of Cilleruelo who studied the case $n=1$.
Donald W Chakeres - One of the best experts on this subject based on the ideXlab platform.
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the harmonic neutron hypothesis prime number factor patterns and their relationship to the hierarchy of the fundamental particles and bosons
Advances in Pure Mathematics, 2015Co-Authors: Donald W Chakeres, Richard VentoAbstract:The Harmonic Neutron Hypothesis, HNH, has demonstrated that many of the fundamental physical constants including particles and bosons are associated with specific quantum Integers, n. These Integers define partial harmonic fractional exponents, 1 ± (1/n), of a fundamental frequency, Vf. The goal is to evaluate the prime and composite factors associated with the neutron n0, the quarks, the kinetic energy of neutron beta decay, the Rydberg constant, R, e, a0, H0, h, α, W, Z, the muon, and the neutron gluon. Their pure number characteristics correspond and explain the hierarchy of the particles and bosons. The elements and black body radiation represent Consecutive Integer series. The relative scale of the constants cluster in a partial harmonic fraction pattern around the neutron. The global numerical organization is related to the only possible prime factor partial fractions of 2/3, or 3/2, as pairs of 3 physical entities with a total of 6 in each group. Many other progressively resonant prime number factor patterns are identified with increasing numbers of smaller factors, higher primes, or larger partial fractions associated with higher order particles or bosons.
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the harmonic neutron hypothesis derivation of planck time and the newtonian constant of gravity from the subatomic properties of a neutron and hydrogen
Particle Physics Insights, 2011Co-Authors: Donald W ChakeresAbstract:Planck time (t P ) is derived from subatomic physical constants: frequency equivalents of the neutron, the electron, the Bohr radius, and the ionization energy of hydrogen. t P squared represents a proportionality constant where the product with the frequency equivalents of two masses and the frequency equivalent distance equals the gravitational energy in Hz. This method is based on the harmonic neutron hypothesis explained herein: the fundamental constants represent a unified exponential Consecutive Integer (forces) or Integer quantum fraction (1 ± 1/n) (particles, bosons, distances) system where the annihilation frequency of the neutron (v n s) is the base. All of the fundamental constants are associated with simple linear relationships of their components when plotted on a ln ln plane using the slopes and intercepts of two ln ln plotted lines associated with hydrogen, weak kinetic, wk, and electromagnetic, em. The degenerate, approximate value of t P 2 can be derived utilizing the quantum fraction values for the proton, 1, gravitational binding energy of electron, −1, the electron, 6/7, and the Bohr radius, 4/5. The approximate degenerate value yielded of t P is the square root of v n raised to the exponent −1−1−6/7−4/5 divided by 4π is 5.51548 × 10 −44 s, and the known value is 5.39124 × 10 −44 s. The predicted degener- ate value of Newton's gravitation constant G is 6.9854466 × 10 −11 m 3 kg −1 s −2 , whereas its known value is 6.67428 × 10 −11 m 3 kg −1 s −2 . Using the hydrogen line values a more precise prediction can be made beyond what can be measured. Two points define the t P 2 line, (0, −b wk −b em ) and (−1, −a wk ). The intercept of this line at the sum of the quantum fractions (−128/35 −1) representing t P 2 is used to derive t P. The hydrogen line derived t P value is 5.391141 × 10 −44 s. The hydrogen line derived G value is 6.6740402 × 10 −11 m 3 kg −1 s −2 . These derived values are within the known uncertainties. This method bridges from subatomic properties of hydrogen to gravity unifying these two systems and multiple forces.
Alexia S. Mintos - One of the best experts on this subject based on the ideXlab platform.
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ON Consecutive Integer PAIRS WITH THE SAME SUM OF DISTINCT PRIME DIVISORS
Integers, 2005Co-Authors: Douglas E. Iannucci, Alexia S. MintosAbstract:We define the arithmetic function P by P (1) = 0, and P (n )= p1 + p2 + ··· + pk if n has the unique prime factorization given by n = k=1 p a i i ; we also define ω(n )= k and ω(1) = 0. We study pairs (n, n + 1) of Consecutive Integers such that P (n )= P (n + 1). We prove that (5, 6), (24, 25), and (49, 50) are the only such pairs (n, n +1 ) where{ω(n) ,ω (n +1 )} = {1, 2} .W e also show how to generate certain pairs of the form (2 2n pq, rs), with p
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on Consecutive Integer pairs with the same sum of distinct prime divisors
Integers, 2005Co-Authors: Douglas E. Iannucci, Alexia S. MintosAbstract:We define the arithmetic function P by P (1) = 0, and P (n )= p1 + p2 + ··· + pk if n has the unique prime factorization given by n = k=1 p a i i ; we also define ω(n )= k and ω(1) = 0. We study pairs (n, n + 1) of Consecutive Integers such that P (n )= P (n + 1). We prove that (5, 6), (24, 25), and (49, 50) are the only such pairs (n, n +1 ) where{ω(n) ,ω (n +1 )} = {1, 2} .W e also show how to generate certain pairs of the form (2 2n pq, rs), with p
Stephan Baier - One of the best experts on this subject based on the ideXlab platform.
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prime powers dividing products of Consecutive Integer values of x 2 n 1 x2n 1
Research in Number Theory, 2020Co-Authors: Stephan Baier, Pallab Kanti DeyAbstract:Let n be a positive Integer and $$f(x) := x^{2^n}+1$$. In this paper, we study orders of primes dividing products of the form $$P_{m,n}:=f(1)f(2)\ldots f(m)$$. We prove that if $$m > \max \{10^{12},4^{n+1}\}$$, then there exists a prime divisor p of $$P_{m,n}$$ such that $$\mathrm{ord}_{p}(P_{m,n} )\le n\cdot 2^{n-1}$$. For $$n=2$$, we establish that for every positive Integer m, there exists a prime divisor p of $$P_{m,2}$$ such that $$\mathrm{ord}_{p} (P_{m,2}) \le 4$$. Consequently, $$P_{m,2}$$ is never a fifth or higher power. This extends work of Cilleruelo [6] who studied the case $$n=1$$.
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prime powers dividing products of Consecutive Integer values of x 2 n 1
arXiv: Number Theory, 2019Co-Authors: Stephan Baier, Pallab Kanti DeyAbstract:Let $n$ be a positive Integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)\cdots f(m)$. We prove that if $m > \max\{10^{12},4^{n+1}\}$, then there exists a prime divisor $p$ of $P_{m,n}$ such that ord$_{p}(P_{m,n} )\leq n\cdot 2^{n-1}$. For $n=2$, we establish that for every positive Integer $m$, there exists a prime divisor $p$ of $P_{m,2}$ such that ord$_{p} (P_{m,2}) \leq 4$. Consequently, $P_{m,2}$ is never a fifth or higher power. This extends work of Cilleruelo who studied the case $n=1$.
Douglas E. Iannucci - One of the best experts on this subject based on the ideXlab platform.
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ON Consecutive Integer PAIRS WITH THE SAME SUM OF DISTINCT PRIME DIVISORS
Integers, 2005Co-Authors: Douglas E. Iannucci, Alexia S. MintosAbstract:We define the arithmetic function P by P (1) = 0, and P (n )= p1 + p2 + ··· + pk if n has the unique prime factorization given by n = k=1 p a i i ; we also define ω(n )= k and ω(1) = 0. We study pairs (n, n + 1) of Consecutive Integers such that P (n )= P (n + 1). We prove that (5, 6), (24, 25), and (49, 50) are the only such pairs (n, n +1 ) where{ω(n) ,ω (n +1 )} = {1, 2} .W e also show how to generate certain pairs of the form (2 2n pq, rs), with p
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on Consecutive Integer pairs with the same sum of distinct prime divisors
Integers, 2005Co-Authors: Douglas E. Iannucci, Alexia S. MintosAbstract:We define the arithmetic function P by P (1) = 0, and P (n )= p1 + p2 + ··· + pk if n has the unique prime factorization given by n = k=1 p a i i ; we also define ω(n )= k and ω(1) = 0. We study pairs (n, n + 1) of Consecutive Integers such that P (n )= P (n + 1). We prove that (5, 6), (24, 25), and (49, 50) are the only such pairs (n, n +1 ) where{ω(n) ,ω (n +1 )} = {1, 2} .W e also show how to generate certain pairs of the form (2 2n pq, rs), with p