New finding of number theory By Liu Ran Contents 1
... If the distance from infinite becomes more and more small, when d(n) = ( ¥ - n) < e , e is smaller than any number, it means it can be smaller than 1. If e <1, then ( ¥ - n) < e <1, Þ n+1 > ¥ , Þ natural number n+1 has exceeded infinite. So supposition is false and natural number is finite. 6. Prime ...
... If the distance from infinite becomes more and more small, when d(n) = ( ¥ - n) < e , e is smaller than any number, it means it can be smaller than 1. If e <1, then ( ¥ - n) < e <1, Þ n+1 > ¥ , Þ natural number n+1 has exceeded infinite. So supposition is false and natural number is finite. 6. Prime ...
New conjectures in number theory
... like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing ...
... like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing ...
Regular Sequences of Symmetric Polynomials
... PROOF. (a) As in the proof of the lemma above, the polynomials pa (2) with a 2 A have a non-trivial common zero if and only if the polynomials pa (2) with a 2 A0 have a non-trivial common zero. So we may assume that A A0 . If all the elements of A are odd then (1; 1) is a non-trivial common zero. ...
... PROOF. (a) As in the proof of the lemma above, the polynomials pa (2) with a 2 A have a non-trivial common zero if and only if the polynomials pa (2) with a 2 A0 have a non-trivial common zero. So we may assume that A A0 . If all the elements of A are odd then (1; 1) is a non-trivial common zero. ...
Number Theory Notes
... Division in modular arithmetic and Euclid’s algorithm So far, we have shown how we can multiply and add in modular arithmetic. We can subtract as well, by combining these two rules: a − b = a + (−1) ∗ b ≡ a + (−1) ∗ b ≡ a − b (mod n), which in hindsight was rather obvious. The next obvious step is t ...
... Division in modular arithmetic and Euclid’s algorithm So far, we have shown how we can multiply and add in modular arithmetic. We can subtract as well, by combining these two rules: a − b = a + (−1) ∗ b ≡ a + (−1) ∗ b ≡ a − b (mod n), which in hindsight was rather obvious. The next obvious step is t ...
The Period and the Distribution of the Fibonacci
... By going backward in this way we can find the smallest natural number u such that u < t, apu+1 = 1 and apu+2 = 1. Therefore Bp ( mod m) must has the period. If we denote the period by kp(m), then kp(m) = u × p. Clearly kp(m) < p × m2 . Remark 3.1. By Theorem 5 the period of Bp is p×s when Bps+1 = Bps ...
... By going backward in this way we can find the smallest natural number u such that u < t, apu+1 = 1 and apu+2 = 1. Therefore Bp ( mod m) must has the period. If we denote the period by kp(m), then kp(m) = u × p. Clearly kp(m) < p × m2 . Remark 3.1. By Theorem 5 the period of Bp is p×s when Bps+1 = Bps ...
Rank statistics for a family of elliptic curves over a function field
... Rq (d) = (log d)(1+o(1)) log log log d for almost all numbers d ∈ Up in the sense of asymptotic density. We hope to take this up in a future paper. Perhaps more importantly, it should be interesting to investigate the situation for more families of elliptic curves than the one family of Ulmer that w ...
... Rq (d) = (log d)(1+o(1)) log log log d for almost all numbers d ∈ Up in the sense of asymptotic density. We hope to take this up in a future paper. Perhaps more importantly, it should be interesting to investigate the situation for more families of elliptic curves than the one family of Ulmer that w ...
Document
... • We will adopt the viewpoint that A number is the root of some polynomial (with integer coefficients). • For example: – “2” is a number which satisfies x2 – 2=0. – “3/4” is a number which satisfies 4x – 3=0. – “10” is a number which satisfies x – 10=0 More precisely, the above numbers are “algebra ...
... • We will adopt the viewpoint that A number is the root of some polynomial (with integer coefficients). • For example: – “2” is a number which satisfies x2 – 2=0. – “3/4” is a number which satisfies 4x – 3=0. – “10” is a number which satisfies x – 10=0 More precisely, the above numbers are “algebra ...