scholarly journals Gaussian primes in narrow sectors

Mathematika ◽  
2001 ◽  
Vol 48 (1-2) ◽  
pp. 119-135 ◽  
Author(s):  
Glyn Harman ◽  
Philip Lewis
Keyword(s):  
Gaussian Primes

scholarly journals A Refined Conjecture for the Variance of Gaussian Primes across Sectors

2020 ◽  
pp. 1-21
Author(s):  
Ryan C. Chen ◽  
Yujin H. Kim ◽  
Jared D. Lichtman ◽  
Steven J. Miller ◽  
Alina Shubina ◽  
...  
Keyword(s):  
Gaussian Primes

The sum of digits of Gaussian primes

2011 ◽  
Vol 27 (1) ◽  
pp. 43-70 ◽  
Author(s):  
Johannes F. Morgenbesser
Keyword(s):  
Sum Of Digits ◽  
Gaussian Primes

On the Gaussian Primes on the Line IM(X) = 1

1973 ◽  
Vol 27 (122) ◽  
pp. 399 ◽  
Author(s):  
M. C. Wunderlich
Keyword(s):  
Gaussian Primes

scholarly journals A conjecture of Paul Erdo’s concerning Gaussian primes

1970 ◽  
Vol 24 (109) ◽  
pp. 221-221
Author(s):  
J. H. Jordan ◽  
J. R. Rabung
Keyword(s):  
Gaussian Primes

Quasi-uniqueness of the set of "Gaussian prime plus one's"

2014 ◽  
Vol 10 (07) ◽  
pp. 1783-1790
Author(s):  
Jay Mehta ◽  
G. K. Viswanadham
Keyword(s):  
Acta Arith ◽  
Gaussian Integers ◽  
Arithmetical Functions ◽  
Set Of Uniqueness ◽  
Positive Integers ◽  
Complex Valued ◽  
Gaussian Primes

We recall the well-known notion of the set of uniqueness for arithmetical functions, introduced by Kátai and several other mathematicians like Indlekofer, Elliot and Hoffman, independently. We define its analogue for completely additive complex-valued functions over the set of non-zero Gaussian integers with some examples. We show that the set of "Gaussian prime plus one's" along with finitely many Gaussian primes of norm up to some constant K is a set of uniqueness with respect to Gaussian integers. This is analogous to Kátai's result in the case of positive integers [I. Kátai, On sets characterizing number theoretical functions, II, Acta Arith.16 (1968) 1–14].

scholarly journals Bounded gaps between Gaussian primes

2017 ◽  
Vol 171 ◽  
pp. 449-473 ◽  
Author(s):  
Akshaa Vatwani
Keyword(s):  
Bounded Gaps ◽  
Gaussian Primes

scholarly journals Mathematical Attack of RSA by Extending the Sum of Squares of Primes to Factorize a Semi-Prime

2020 ◽  
Vol 25 (4) ◽  
pp. 63
Author(s):  
Anthony Overmars ◽  
Sitalakshmi Venkatraman
Keyword(s):  
Cycle Time ◽  
Factorization Method ◽  
Sum Of Squares ◽  
Sums Of Squares ◽  
Arithmetic Operations ◽  
Search Field ◽  
Sums Of Two Squares ◽  
Factorization Algorithms ◽  
Gaussian Primes

The security of RSA relies on the computationally challenging factorization of RSA modulus N=p1 p2 with N being a large semi-prime consisting of two primes p1and p2, for the generation of RSA keys in commonly adopted cryptosystems. The property of p1 and p2, both congruent to 1 mod 4, is used in Euler’s factorization method to theoretically factorize them. While this caters to only a quarter of the possible combinations of primes, the rest of the combinations congruent to 3 mod 4 can be found by extending the method using Gaussian primes. However, based on Pythagorean primes that are applied in RSA, the semi-prime has only two sums of two squares in the range of possible squares N−1, N/2 . As N becomes large, the probability of finding the two sums of two squares becomes computationally intractable in the practical world. In this paper, we apply Pythagorean primes to explore how the number of sums of two squares in the search field can be increased thereby increasing the likelihood that a sum of two squares can be found. Once two such sums of squares are found, even though many may exist, we show that it is sufficient to only find two solutions to factorize the original semi-prime. We present the algorithm showing the simplicity of steps that use rudimentary arithmetic operations requiring minimal memory, with search cycle time being a factor for very large semi-primes, which can be contained. We demonstrate the correctness of our approach with practical illustrations for breaking RSA keys. Our enhanced factorization method is an improvement on our previous work with results compared to other factorization algorithms and continues to be an ongoing area of our research.

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