Add to ORKGDenote by ρ(k) the smallest prime number with digital sum k (not a multiple of 3). Richard K. Guy asked whether the congruences ρ(k) ≡ 99 (mod 100) and ρ(k) ≡ 999 (mod 1000) hold for all k> 38, respectively k> 59. Counterexamples to this are given, inter alia, for k = 86 and k = 104. Moreover, several open problems and conjectures about ρ(k) are discussed. 1
AbstractLet p be an odd prime and γ(k,pn) be the smallest positive integer s such that every integer...
Abstract This paper builds on Goldbach’s weak conjecture, showing that all primes to infinity are co...
Programs due to Wirth and Misra for generating the prime numbers up to a specified limit are invest...
Denote by ρ(k) the smallest prime number with digital sum k (not a multiple of 3). Richard K. Guy as...
This paper describes the construction of the first explicitly known widely digitally delicate prime....
Abstract. Let p be an odd prime and γ(k, pn) be the smallest positive integer s such that every inte...
A finite ncreasing sequence {pn}, (n=1, 2, 3, ・・・, t) of prime numbers, (t〓3), is called an arithmet...
Using Fermat’s Little Theorem, it can be shown that Σmi=1 i m−1 ≡ −1 (mod m) if m is prime. It has b...
For any k ≥ 0, all primes n satisfy the congruence nσk(n) ≡ 2 mod ϕ(n). We show that this congruenc...
We study integers n > 1 satisfying the relation σ(n) = γ(n)², where σ(n) and γ(n) are the sum of div...
AbstractMany people are fascinated by π Vast amounts of human and computer resourceshave been spent ...
. Let p be a prime congruent to \Gamma1 modulo 4, i n p j the Legendre symbol and S(k) = P p\Ga...
Let kα be the least positive integer such that 2 α kα is not a value of Euler’s phi-function. In the...
We study integers n > 1 satisfying the relation σ(n) = γ(n) ² , where σ(n) and γ(n) are the sum of d...
AbstractIn this paper, we are able to sharpen Hua's classical result by showing that each sufficient...
AbstractLet p be an odd prime and γ(k,pn) be the smallest positive integer s such that every integer...
Abstract This paper builds on Goldbach’s weak conjecture, showing that all primes to infinity are co...
Programs due to Wirth and Misra for generating the prime numbers up to a specified limit are invest...
Denote by ρ(k) the smallest prime number with digital sum k (not a multiple of 3). Richard K. Guy as...
This paper describes the construction of the first explicitly known widely digitally delicate prime....
Abstract. Let p be an odd prime and γ(k, pn) be the smallest positive integer s such that every inte...
A finite ncreasing sequence {pn}, (n=1, 2, 3, ・・・, t) of prime numbers, (t〓3), is called an arithmet...
Using Fermat’s Little Theorem, it can be shown that Σmi=1 i m−1 ≡ −1 (mod m) if m is prime. It has b...
For any k ≥ 0, all primes n satisfy the congruence nσk(n) ≡ 2 mod ϕ(n). We show that this congruenc...
We study integers n > 1 satisfying the relation σ(n) = γ(n)², where σ(n) and γ(n) are the sum of div...
AbstractMany people are fascinated by π Vast amounts of human and computer resourceshave been spent ...
. Let p be a prime congruent to \Gamma1 modulo 4, i n p j the Legendre symbol and S(k) = P p\Ga...
Let kα be the least positive integer such that 2 α kα is not a value of Euler’s phi-function. In the...
We study integers n > 1 satisfying the relation σ(n) = γ(n) ² , where σ(n) and γ(n) are the sum of d...
AbstractIn this paper, we are able to sharpen Hua's classical result by showing that each sufficient...
AbstractLet p be an odd prime and γ(k,pn) be the smallest positive integer s such that every integer...
Abstract This paper builds on Goldbach’s weak conjecture, showing that all primes to infinity are co...
Programs due to Wirth and Misra for generating the prime numbers up to a specified limit are invest...