This paper deals with the period T of pseudorandom number sequence gener- ated by subtract- with borrow (SWB ). For the selected base b,legs r > s it is shown that T is dependent only on b, r and s and equal to the...This paper deals with the period T of pseudorandom number sequence gener- ated by subtract- with borrow (SWB ). For the selected base b,legs r > s it is shown that T is dependent only on b, r and s and equal to the order of b with respect to m = br- bs + 1 provided that m and A, which is determined by initial values x1,s’’’ ,xr and initial borrow value cr+1, are coprime. For b = 231-1 this paper searches the field 0 < r ≤ 300, r-s ≤ 10 and finds out all the pairs of (r, s) for which m is prime. In addition, the periods of SWB corresponding to b = 231-1 and these pairs (r, s) are aIso calculated. Besed on data obtained, generators of SWB with extremely long period could be designed.展开更多
This research work relates the surface of a square and the area circumscribed by a circle, resulting in a value called Nikola Tesla constant. This constant starts with the calculation of the areas of the square and th...This research work relates the surface of a square and the area circumscribed by a circle, resulting in a value called Nikola Tesla constant. This constant starts with the calculation of the areas of the square and the inscribed circle, giving a ratio of 9/40 and from which a residual area of the area proportions of the geometric figures described is obtained. Plotting smooth curves, particularly those in round shapes, can be represented efficiently with the use of Nikola Tesla constant, reducing complex mathematical calculus.展开更多
This work presents a different approach to twin primes, an approach from the perspective of the Tesla numbers and gives a refresh and new observation of twin primes that could lead us to an answer to the Twin Prime Co...This work presents a different approach to twin primes, an approach from the perspective of the Tesla numbers and gives a refresh and new observation of twin primes that could lead us to an answer to the Twin Prime Conjecture problem. We expose a peculiar relation between twin primes and the generation of prime numbers with Tesla numbers. Tesla numbers seem to be present in so many domains like time, vibration and frequency [1], and the space between twin primes is not the exception. Let us say that twin primes are more than just prime numbers plus 2 or minus 2, and Tesla numbers are more involved with twin primes than we think, and hopefully, this approach give us a better understanding of the distribution of the twin pairs.展开更多
Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand even to the case of ax2+bxy+cy2=n. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equa...Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand even to the case of ax2+bxy+cy2=n. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equation x2+y2=n). Starting from a quadratic form with two variables f(x,y)=ax2+bxy+cy2and n an integer. We have shown that a primitive positive solution (u,v)of the equation f(x,y)=nis admissible if it is obtained in the following way: we take α modulo n such that f(α,1)≡0modn, u is the first of the remainders of Euclid’s algorithm associated with n and α that is less than 4cn/| D |) (possibly α itself) and the equation f(x,y)=n. has an integer solution u in y. At the end of our work, it also appears that the Cornacchia algorithm is good for the form n=ax2+bxy+cy2if all the primitive positive integer solutions of the equation f(x,y)=nare admissible, i.e. computable by the algorithmic process.展开更多
文摘This paper deals with the period T of pseudorandom number sequence gener- ated by subtract- with borrow (SWB ). For the selected base b,legs r > s it is shown that T is dependent only on b, r and s and equal to the order of b with respect to m = br- bs + 1 provided that m and A, which is determined by initial values x1,s’’’ ,xr and initial borrow value cr+1, are coprime. For b = 231-1 this paper searches the field 0 < r ≤ 300, r-s ≤ 10 and finds out all the pairs of (r, s) for which m is prime. In addition, the periods of SWB corresponding to b = 231-1 and these pairs (r, s) are aIso calculated. Besed on data obtained, generators of SWB with extremely long period could be designed.
文摘研究了一类不定方程求正整数解的问题.借助一个引理,推导并证明了不定方程x2-py2=z2(p为奇素数)正整数解的一般公式.不定方程x2-py2=z2(p为奇素数)满足(x,y)=1的一切正整数解可表示为x=12(a2+pb2),y=ab,z=12a2-pb2,这里a>0,b>0,a,b都是奇数,p a;或x=a2+pb2,y=2ab,z=a2-pb2,这里a>0,b>0,a,b一奇一偶,p a.
文摘This research work relates the surface of a square and the area circumscribed by a circle, resulting in a value called Nikola Tesla constant. This constant starts with the calculation of the areas of the square and the inscribed circle, giving a ratio of 9/40 and from which a residual area of the area proportions of the geometric figures described is obtained. Plotting smooth curves, particularly those in round shapes, can be represented efficiently with the use of Nikola Tesla constant, reducing complex mathematical calculus.
文摘This work presents a different approach to twin primes, an approach from the perspective of the Tesla numbers and gives a refresh and new observation of twin primes that could lead us to an answer to the Twin Prime Conjecture problem. We expose a peculiar relation between twin primes and the generation of prime numbers with Tesla numbers. Tesla numbers seem to be present in so many domains like time, vibration and frequency [1], and the space between twin primes is not the exception. Let us say that twin primes are more than just prime numbers plus 2 or minus 2, and Tesla numbers are more involved with twin primes than we think, and hopefully, this approach give us a better understanding of the distribution of the twin pairs.
文摘Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand even to the case of ax2+bxy+cy2=n. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equation x2+y2=n). Starting from a quadratic form with two variables f(x,y)=ax2+bxy+cy2and n an integer. We have shown that a primitive positive solution (u,v)of the equation f(x,y)=nis admissible if it is obtained in the following way: we take α modulo n such that f(α,1)≡0modn, u is the first of the remainders of Euclid’s algorithm associated with n and α that is less than 4cn/| D |) (possibly α itself) and the equation f(x,y)=n. has an integer solution u in y. At the end of our work, it also appears that the Cornacchia algorithm is good for the form n=ax2+bxy+cy2if all the primitive positive integer solutions of the equation f(x,y)=nare admissible, i.e. computable by the algorithmic process.
