scholarly journals Twin Prime Conjecture - Proof that there are infinite twin primes.

2021 ◽  
Author(s):  
Vijay Nandanwar
Keyword(s):  
Lower Bound ◽  
Number Series ◽  
Twin Primes

We determine that twin primes are not random within specific range of number series. Within the range, they are part of an infinite repeating cycle within the series. The repeating cycles of this series is symmetrical and that range is part of one such cycle. We can also determine where in the repeating cycle this range is located. Once we establish the repeating cycles, we find the probability of twin primes for the cycle/series. Next we determine the lower bound for the probability and the range. Finally, we prove that the twin prime conjecture is true.

scholarly journals The Brocard conjecture

2021 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Lower Bound ◽  
Great Degree ◽  
Common Ground ◽  
Prime Numbers ◽  
Upper Bounds ◽  
Lower Upper ◽  
Twin Primes ◽  
True False

The Brocard conjecture asserts that the number of primes, within the interval, between the squares of two subsequent primes is greater than or equal to 4. Although the number of primes within this interval varies to a great degree, there is a common ground, which makes it possible to settle this old conundrum. Three bounds are developed: the least lower bound and the lower/upper bounds. The least lower bound is implemented to prove the conjecture. The lower/upper bounds exploit the shortest such interval, namely between the twin primes. This has been done in order to establish the bounds, on the smallest number of primes within that interval. The research objective was not only to provide a true/false answer, but to clarify some aspects of the distribution of prime numbers within this interval as well.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

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