By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or ...By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2p + 1 divides the Mersenne number Mp. The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents Mp.展开更多
When we study a congruence T(x) ≡ ax modulo m as pseudo random number generator, there are several means of ensuring the independence of two successive numbers. In this report, we show that the dependence depends on ...When we study a congruence T(x) ≡ ax modulo m as pseudo random number generator, there are several means of ensuring the independence of two successive numbers. In this report, we show that the dependence depends on the continued fraction expansion of m/a. We deduce that the congruences such that m and a are two successive elements of Fibonacci sequences are those having the weakest dependence. We will use this result to obtain truly random number sequences xn. For that purpose, we will use non-deterministic sequences yn. They are transformed using Fibonacci congruences and we will get by this way sequences xn. These sequences xn admit the IID model for correct model.展开更多
Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizatio...Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizations have been studied intensively. In this note, we consider the congruences involving the combination of alternating harmonic sums, <img alt="" src="Edit_e97d0c64-3683-4a75-9d26-4b371c2be41e.bmp" /> where P<em><sub>P </sub></em>denotes the set of positive integers which are prime to <em>p</em>. And we establish the combinational congruences involving alternating harmonic sums for positive integer <em>n</em>=3,4,5.展开更多
An elementary formula to know the number of primes in the interval (x, 2x) close to the exact figure for a fixed x is given here. A new elementary equation is derived (a relation between prime numbers and composite nu...An elementary formula to know the number of primes in the interval (x, 2x) close to the exact figure for a fixed x is given here. A new elementary equation is derived (a relation between prime numbers and composite numbers distributed in the interval [1, 2x]). An elementary method to know the number of primes in a given magnitude is suitably placed in the form of a general formula, and we have proved it. The general formula is applied to the terms of the equation, and a tactical simplification of the terms gives rise to an expression whose verification envisages scope for its further studies.展开更多
This study describes how one can construct sets of composite natural numbers as tensorial products of the vectors created with the natural powers of prime numbers.
Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and...Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that ∑k=0^p-1（k^2k/2k）≡（-1）^（p-1）/2-p^2Ep-3（modp^3） ∑k=1^（p-1）/2（k^2k）/k≡（-1）^（p＋1）/2 8/3pEp-3（mod p^2）,∑k=0^（p-1）/2（k^2k）^2/16k≡（-1）^（p-1）/2＋p^2Ep-3（mod p^3）,where E0, E1, E2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for π2, π-2 and the constant K ：= ∑k=1^∞（k/3）/k^2 （with （-） the Jacobi symbol）, two of which are ∑k=1^∞（10k-3）8k/k2（k^2k）^2（k^3k）=π^2/2and ∑k=1^∞（15k-4）（-27）^k-1/k^3（k^2k）^2（k^3k）=K.展开更多
The conjecture of twin prime numbers is a mathematical problem. Proving the twin prime conjecture using traditional modern number theory is extremely profound and complex. We propose an elementary research method for ...The conjecture of twin prime numbers is a mathematical problem. Proving the twin prime conjecture using traditional modern number theory is extremely profound and complex. We propose an elementary research method for corresponding prime number, proved that the conjecture of twin prime numbers and obtain the corresponding prime distribution equation. According to the distribution rate of corresponding prime numbers, the distribution pattern of twin prime numbers was proved the distribution rate theorem. This is the distribution rate of prime numbers corresponding to composite numbers, which approaches the distribution rate of prime numbers corresponding to integers. Based on the corresponding prime distribution equation, obtain the twin prime inequality function. Then, the formula for calculating twin prime numbers was discussed. There is also the Hardy Littlewood conjecture. This provides a practical and feasible approach for studying the distribution of twin prime numbers.展开更多
Based on the data(including radius of maximum winds) from the JTWC(Joint Typhoon Warning Center),the tropical cyclones(TCs) radii of the outermost closed isobar, TCs best tracks from Shanghai Typhoon Institute and the...Based on the data(including radius of maximum winds) from the JTWC(Joint Typhoon Warning Center),the tropical cyclones(TCs) radii of the outermost closed isobar, TCs best tracks from Shanghai Typhoon Institute and the Black Body Temperature(TBB) of the Japanese geostationary meteorological satellite M1 TR IR1, and combining13 tropical cyclones which landed in China again after visiting the island of Taiwan during the period from 2001 to2010, we analyzed the relationship between the number of convective cores within TC circulation and the intensity of TC with the method of convective-stratiform technique(CST) and statistical and composite analysis. The results are shown as follows:(1) The number of convective cores in the entire TC circulation is well corresponding with the outer spiral rainbands and the density of convective cores in the inner core area increases(decreases) generally with increasing(decreasing) TC intensity. At the same time, the number of convective cores within the outer spiral rainbands is more than that within the inner core and does not change much with the TC intensity. However, the density of convective cores within the outer spiral rainbands is lower than that within the inner core.(2) The relationship described above is sensitive to landing location to some extent but not sensitive to the structure of TC.(3) The average value of TBB in the inner core area increases(decreases) generally with increasing(decreasing) of TC intensity, which is also sensitive to landing situation to some extent. At the same time, the average value of TBB within the outer spiral rainbands is close to that within the entire TC circulation, and both of them are more than that within the inner core. However, they do not reflect TC intensity change significantly.(4) The results of statistical composite based on convective cores and TBB are complementary with each other, so a combination of both can reflect the relationship between TC rainbands and TC intensity much better.展开更多
Recently Hong Shaofang[6] has investigated the sums (np + j)-r ( with an odd prime number p 5 and n, r N) by Washington’s p-adic expansion of these sums as a power series in n where the coefficients are values of p-a...Recently Hong Shaofang[6] has investigated the sums (np + j)-r ( with an odd prime number p 5 and n, r N) by Washington’s p-adic expansion of these sums as a power series in n where the coefficients are values of p-adic L-fuctions[12]. Herethe author shows how a more general sums (npl +j)-r,l N, may be studied by elementary methods.展开更多
Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then...Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then (modp^2). (iii)-(2n+1)p (modp^2). This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolstenholme’s theorem is obtained.展开更多
We prove some congruence relations satisfied by integral Stirling type pairs. The results settle a question posed by Hsu . In particular, they extend known congruence properties of Stirling numbers of the first kind ...We prove some congruence relations satisfied by integral Stirling type pairs. The results settle a question posed by Hsu . In particular, they extend known congruence properties of Stirling numbers of the first kind and the second kind.展开更多
This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together wi...This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.展开更多
文摘By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2p + 1 divides the Mersenne number Mp. The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents Mp.
文摘When we study a congruence T(x) ≡ ax modulo m as pseudo random number generator, there are several means of ensuring the independence of two successive numbers. In this report, we show that the dependence depends on the continued fraction expansion of m/a. We deduce that the congruences such that m and a are two successive elements of Fibonacci sequences are those having the weakest dependence. We will use this result to obtain truly random number sequences xn. For that purpose, we will use non-deterministic sequences yn. They are transformed using Fibonacci congruences and we will get by this way sequences xn. These sequences xn admit the IID model for correct model.
文摘Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizations have been studied intensively. In this note, we consider the congruences involving the combination of alternating harmonic sums, <img alt="" src="Edit_e97d0c64-3683-4a75-9d26-4b371c2be41e.bmp" /> where P<em><sub>P </sub></em>denotes the set of positive integers which are prime to <em>p</em>. And we establish the combinational congruences involving alternating harmonic sums for positive integer <em>n</em>=3,4,5.
文摘An elementary formula to know the number of primes in the interval (x, 2x) close to the exact figure for a fixed x is given here. A new elementary equation is derived (a relation between prime numbers and composite numbers distributed in the interval [1, 2x]). An elementary method to know the number of primes in a given magnitude is suitably placed in the form of a general formula, and we have proved it. The general formula is applied to the terms of the equation, and a tactical simplification of the terms gives rise to an expression whose verification envisages scope for its further studies.
文摘This study describes how one can construct sets of composite natural numbers as tensorial products of the vectors created with the natural powers of prime numbers.
基金supported by the National Natural Science Foundation of China(GrantNo.10871087)the Overseas Cooperation Fund of China(Grant No.10928101)
文摘Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that ∑k=0^p-1（k^2k/2k）≡（-1）^（p-1）/2-p^2Ep-3（modp^3） ∑k=1^（p-1）/2（k^2k）/k≡（-1）^（p＋1）/2 8/3pEp-3（mod p^2）,∑k=0^（p-1）/2（k^2k）^2/16k≡（-1）^（p-1）/2＋p^2Ep-3（mod p^3）,where E0, E1, E2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for π2, π-2 and the constant K ：= ∑k=1^∞（k/3）/k^2 （with （-） the Jacobi symbol）, two of which are ∑k=1^∞（10k-3）8k/k2（k^2k）^2（k^3k）=π^2/2and ∑k=1^∞（15k-4）（-27）^k-1/k^3（k^2k）^2（k^3k）=K.
文摘The conjecture of twin prime numbers is a mathematical problem. Proving the twin prime conjecture using traditional modern number theory is extremely profound and complex. We propose an elementary research method for corresponding prime number, proved that the conjecture of twin prime numbers and obtain the corresponding prime distribution equation. According to the distribution rate of corresponding prime numbers, the distribution pattern of twin prime numbers was proved the distribution rate theorem. This is the distribution rate of prime numbers corresponding to composite numbers, which approaches the distribution rate of prime numbers corresponding to integers. Based on the corresponding prime distribution equation, obtain the twin prime inequality function. Then, the formula for calculating twin prime numbers was discussed. There is also the Hardy Littlewood conjecture. This provides a practical and feasible approach for studying the distribution of twin prime numbers.
基金National Natural Science Foundation of China(NSFC)(40875025,41175050,41475039 and41475041)Shanghai Natural Science Foundation of China(08ZR1422900)Public Sector(Meteorology)Research of China(GYHY201306012)
文摘Based on the data(including radius of maximum winds) from the JTWC(Joint Typhoon Warning Center),the tropical cyclones(TCs) radii of the outermost closed isobar, TCs best tracks from Shanghai Typhoon Institute and the Black Body Temperature(TBB) of the Japanese geostationary meteorological satellite M1 TR IR1, and combining13 tropical cyclones which landed in China again after visiting the island of Taiwan during the period from 2001 to2010, we analyzed the relationship between the number of convective cores within TC circulation and the intensity of TC with the method of convective-stratiform technique(CST) and statistical and composite analysis. The results are shown as follows:(1) The number of convective cores in the entire TC circulation is well corresponding with the outer spiral rainbands and the density of convective cores in the inner core area increases(decreases) generally with increasing(decreasing) TC intensity. At the same time, the number of convective cores within the outer spiral rainbands is more than that within the inner core and does not change much with the TC intensity. However, the density of convective cores within the outer spiral rainbands is lower than that within the inner core.(2) The relationship described above is sensitive to landing location to some extent but not sensitive to the structure of TC.(3) The average value of TBB in the inner core area increases(decreases) generally with increasing(decreasing) of TC intensity, which is also sensitive to landing situation to some extent. At the same time, the average value of TBB within the outer spiral rainbands is close to that within the entire TC circulation, and both of them are more than that within the inner core. However, they do not reflect TC intensity change significantly.(4) The results of statistical composite based on convective cores and TBB are complementary with each other, so a combination of both can reflect the relationship between TC rainbands and TC intensity much better.
文摘Recently Hong Shaofang[6] has investigated the sums (np + j)-r ( with an odd prime number p 5 and n, r N) by Washington’s p-adic expansion of these sums as a power series in n where the coefficients are values of p-adic L-fuctions[12]. Herethe author shows how a more general sums (npl +j)-r,l N, may be studied by elementary methods.
文摘Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then (modp^2). (iii)-(2n+1)p (modp^2). This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolstenholme’s theorem is obtained.
文摘We prove some congruence relations satisfied by integral Stirling type pairs. The results settle a question posed by Hsu . In particular, they extend known congruence properties of Stirling numbers of the first kind and the second kind.
文摘This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.
