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On the largest prime factor of $x^{2}-1$

Florian Luca, Filip Najman
2010 Mathematics of Computation  
For example, for any positive integer n we can find the largest positive integer x such that all prime factors of each of x, x + 1, . . . , x + n are less than 100.  ...  In this paper, we find all integers x such that x 2 −1 has only prime factors smaller than 100. This gives some interesting numerical corollaries.  ...  The second author was supported by the Ministry of Science, Education and Sports, Republic of Croatia, grant 037-0372781-2821.  ... 
doi:10.1090/s0025-5718-2010-02381-6 fatcat:wfcwceabjbdzfkpmrgsgwlsple

On the largest prime factor of x^2-1 [article]

Florian Luca, Filip Najman
2010 arXiv   pre-print
For example, for any positive integer n we can find the largest positive integer x such that all prime factors of each of x, x+1,..., x+n are less than 100.  ...  In this paper, we find all integers x such that x^2-1 has only prime factors smaller than 100. This gives some interesting numerical corollaries.  ...  Introduction For any integer n we let P (n) be the largest prime factor of n with the convention P (0) = P (±1) = 1.  ... 
arXiv:1005.1533v1 fatcat:cihtnuagybflxkhgamgk6j3oqu

Large strings of consecutive smooth integers

Filip Najman
2011 Archiv der Mathematik  
prime factors exceeds k.  ...  In this note we improve an algorithm from a recent paper by Bauer and Bennett for computing a function of Erdös that measures the minimal gap size f (k) in the sequence of integers at least one of whose  ...  These materials are based on work financed by the National Foundation for Science, Higher Education and Technological Development of the Republic of Croatia.  ... 
doi:10.1007/s00013-011-0301-y fatcat:ilgitntb6raatevcv6r5ht6mxu

Smooth Operator – The Use of Smooth Integers in Fast Generation of RSA Keys [article]

Vassil Dimitrov, Luigi Vigneri, Vidal Attias
2019 arXiv   pre-print
Typically, the algorithms used have two parts - trial divisions aimed at eliminating numbers with small prime factors and primality tests based on an easy-to-compute statement that is valid for primes  ...  The problem has been a subject of deep investigations by the computational number theorists, but there is still room for improvement.  ...  The perfect powers of 2 are 1-integers (their largest prime factor is 2). 2.  ... 
arXiv:1912.11546v1 fatcat:iivew6kslzemngvdtdm5vhfumy

Large strings of consecutive smooth integers [article]

Filip Najman
2011 arXiv   pre-print
prime factors exceeds k.  ...  In this note we improve an algorithm from a recent paper by Bauer and Bennett for computing a function of Erdös that measures the minimal gap size f(k) in the sequence of integers at least one of whose  ...  These materials are based on work financed by the National Foundation for Science, Higher Education and Technological Development of the Republic of Croatia.  ... 
arXiv:1108.3710v1 fatcat:ijtaas4dwrhxlmlkrtj3cgbc6a

Number Theory, Dialogue, and the Use of Spreadsheets in Teacher Education

Sergei Abramovich
2011 Spreadsheets in Education  
It emphasizes both the power and deficiency of inductive reasoning using a number of historically significant examples.  ...  The notion of computational experiment as a modern approach to the teaching of mathematics is discussed.  ...  That is, it shows that 25 is not divisible by 2; finally, 25 turns out to be divisible by 5 with an integer quotient, 5, greater than 1.  ... 
doaj:6828e17dced349ea8f8c3739a83e4616 fatcat:bjtzyydp6bb4rn4umpcb2xliz4

Page 270 of School Science and Mathematics Vol. 24, Issue 3 [page]

1924 School Science and Mathematics  
The theory of numbers is, properly speaking, the study of relationships which exist between integers. All numbers are divided into two main divisions, namely, prime and com- posite.  ...  For instance it can be verified that the number 2% +1,.a number of more than twenty trillion places has the factor 2,748,779,- 069,441, by the method of congruences.  ... 

Prime factors of binomial coefficients and related problems

P. Erdös, C. Lacampagne, J. Selfridge
1988 Acta Arithmetica  
Then one of these and its corresponding a would be divisible by (s + 1) r . But (s + 1) r > k so this a would be greater than k which violates property (i) .  ...  In the general case, consider integers r and s such that kl(s+ 1) < r < kls where s < k/2 . In any sequence of k consecutive integers there are s or s + 1 consecutive multiples of r .  ...  With n divisible by pn(p~a s above, the sequence of consecutive integers associated with the symmetric flip of the identity is n-k, n-k+ 1, . .  ... 
doi:10.4064/aa-49-5-507-523 fatcat:lxrkmyps2rfthbfhretcgqbamu

The Role of Smooth Numbers in Number Theoretic Algorithms [chapter]

Carl Pomerance
1995 Proceedings of the International Congress of Mathematicians  
A proof of the first statement, which is considerably easier than a proof of the second, is implicit in [BLP, Theorem 10.1], and explicit in [P4, Proposition 4.1].  ...  L(xy 2~E integers, the probability tends to 0.  ...  Afford, A. Granville, and H. Lenstra for their helpful critical comments on an earlier draft of this paper.  ... 
doi:10.1007/978-3-0348-9078-6_34 fatcat:zxkfpzrbqzfunlvlrrv3yrdn34

Subsets of an interval whose product is a power

Paul Erdős, Janice L. Malouf, J.L. Selfridge, Esther Szekeres
1999 Discrete Mathematics  
Abstract We form squares from the product of integers in a short interval [n, n + t,], where we include n in the product. If p is prime, pin, and (P) >n, we prove that p is the minimum t".  ...  Correspondingly, the lower bound gk >~(k + 1)p in Conjecture 8 is implied by the conjunction of Conjectures 12 and 13.  ...  Write n = ap where p is the largest prime factor of the square-free part of n. If p > 2a + 1 we obtain 6 integer products which are products of a, a ÷ ½, a + 1 and p-l, p, p+l.  ... 
doi:10.1016/s0012-365x(98)00332-x fatcat:svo6ihxjmrh3fhmevpdm4ijoie

Primitive and geometric-progression-free sets without large gaps [article]

Nathan McNew
2019 arXiv   pre-print
We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound  ...  for the gaps in the sequence of prime numbers.  ...  For those primes y ′ 2 ≤ p < y, we can bound the number of integers in S divisible by p trivially by y p = O(1).  ... 
arXiv:1809.08355v2 fatcat:gq25y7zpr5h6tcpzzafydivngm

Weights Modulo a Prime Power in Divisible Codes and a Related Bound

X. Liu
2006 IEEE Transactions on Information Theory  
Using a similar idea, we give an upper bound for the dimension of a divisible code by some divisibility property of its weight enumerator modulo p e .  ...  In this paper, we generalize the theorem given by R. M. Wilson about weights modulo p t in linear codes to a divisible code version.  ...  A divisible code is a linear code whose codewords all have weights divisible by some integer , where is called a divisor of the code. Let be a prime, and , , be a prime power.  ... 
doi:10.1109/tit.2006.881708 fatcat:vtfn376pqjgphbo6trulrdzeg4

Page 66 of Mathematical Reviews Vol. 3, Issue 3 [page]

1942 Mathematical Reviews  
It is proved that, when m2=17, there exists a set of consecutive integers such that no number in the set is prime to all the rest. The same theorem has been proved by A. Brauer [Bull. Amer. Math.  ...  Czech and German summaries) [MF 5205] Consider the congruence f(x) =x*+a;x*"'+ - --+a,=0 (mod p), where the coefficients are integers, p is a prime and the dis- criminant D is not divisible by p.  ... 

Pairs of consecutive power residues

D. H. Lehmer, Emma Lehmer, W. H. Mills
1963 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
For a fixed k and m we call a prime exceptional and denote it by p* if there do not exist m consecutive integers each of which is a &th power residue of p*.  ...  According to a theorem of Brauer (1), for every sufficiently large prime p there exist m consecutive positive integers r, r + l,...,r + m -1, each of which is a &th power residue of p.  ...  For a fixed k and m we call a prime exceptional and denote it by p* if there do not exist m consecutive integers each of which is a &th power residue of p*.  ... 
doi:10.4153/cjm-1963-020-4 fatcat:6dfcjerhtndv7dy45dirsp7e6q

Page 5904 of Mathematical Reviews Vol. , Issue 93k [page]

1993 Mathematical Reviews  
The authors search out the irregular primes up to 4 million and also note the index of irregularity (the number of Bernoulli numbers B, divisible by the prime p for even ¢ < p — 1).  ...  L. (1-NIL) Estimates of the least prime factor of a binomial coefficient. (English summary) Math. Comp. 61 (1993), no. 203, 215-224.  ... 
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