The greatest prime factor of the integers in a short interval (III)

LetP(x) denote the greatest prime factor of II_z<n≤x+x1/2 n. In this paper, we prove thatP(x)>x ^0.723 holds true for a sufficiently largex. Project supported by the Tian Yuan Itam in the National Natural Science Foundation of China.     

On the greatest prime factor of integers of the form ab + 1

Let N be a positive integer and let A and B be dense subsets of {1,...,N }. The purpose of this paper is to establish a good lower bound for the greatest prime factor of ab + 1as a and b run over the elements of A and B respectively.     

On the greatest prime factor of

We prove that for integers b>c>0$">, the greatest prime factor of tends to infinity with . In particular, this settles a conjecture raised by Györy, Sarkozy and Stewart, predicting the same conclusion for the product .

     

On the greatest prime factor of ab + 1

We prove that whenever $ \mathcal{A} $ and $ \mathcal{B} $ are dense enough subsets of {1, ..., N}, there exist a ∈ $ \mathcal{A} $ and b ∈ $ \mathcal{B} $ such that the greatest prime factor of ab + 1 is at least $ N^{1 + |\mathcal{A}|/(9N)} $ .     

On the greatest prime factor of ab + 1

We improve some results on the size of the greatest prime factor of the integers of the form ab + 1 where a and b belong to some general given finite sequences A and B with rather large density.     

Three primes theorem in a short interval (VII)

In this paper, we shall prove that for a sufficiently large odd numberN, the equation

Three primes theorem in a short interval (V)

In this paper, we shall prove that for a sufficiently large odd numberN, the equation has solutions. The Project Supported by National Natural Science Foundation of China     

The number of large prime divisors of consecutive integers

Let ?(N) > 0 be a function of positive integers N and such that ?(N) → 0 and N^?(N) → ∞ as N → + ∞. Let Nν_N(n:…) be the number of positive integers n ≤ N for which the property stated in the dotted space holds. Finally, let g(n; N, ?, z) be the number of those prime divisors p of n which satisfy N^Z?(N) ? p ? N^?(N), 0 < z < 1 In the present note we show that for each k = 0, ±1, ±2,…, as N → ∞, limv_N(n : g(n; N, ?, z) ? g(n + 1; N, ?z) = k) exists and we determine its actual value. The case k = 0 induced the present investigation. Our solution for this value shows that the natural density of those integers n for which n and n + 1 have the same number of prime divisors in the range (1) exists and it is positive.     

The Titchmarsh problem with integers having a given number of prime divisors

The asymptotics for the number of representations ofN asN→∞ is expressed as the sum of a number havingk prime divisors and a product of two natural numbers. The asymptotics is found fork≤(2−ε) ln lnN and (2+ε) ln lnN≤k≤b ln lnN, whereε>0. The results obtained are uniform with respect tok. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 585–602, April, 1996. This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00260.     

Approximating the number of integers without large prime factors

denotes the number of positive integers and free of prime factors . Hildebrand and Tenenbaum gave a smooth approximation formula for in the range , where is a fixed positive number . In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate . The computational complexity of this algorithm is . We give numerical results which show that this algorithm provides accurate estimates for and is faster than conventional methods such as algorithms exploiting Dickman's function.

     

On the sum of a prime and a k-th power of prime in short intervals

Let H_k\mathcal{H}_{k} denote the set {n∣2|n, n\not o 1 (mod p)n\not\equiv 1\ (\mathrm{mod}\ p) ∀ p>2 with p−1|k}. We prove that when X^{\frac1120(1-\frac12k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n ? \allowbreak H_k ?(X, X+H]n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]} can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when X^{\frac1120(1-\frac1k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n∈(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3.     

Additive problems for integers with a given number of prime divisors

The asymptotics of sums of the form Στ(|bn−a|) (summation overn<N, ω(n)=k) is studied, whereω(n) is the number of distinct prime divisors ofn, andτ(n) is the number of all divisors. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 749–762, May, 1998. In conclusion, the author wishes to express his gratitude to Professor N. M. Timofeev for valuable advice. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00502.     

On sums of a prime and four prime squares in short intervals

In this paper, we are able to sharpen Hua's classical result by showing that each sufficiently large integer can be written as

     

Approximating the number of integers free of large prime factors

Define to be the number of positive integers such that has no prime divisor larger than . We present a simple algorithm that approximates in floating point operations. This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than other known approximations, including the well-known approximation , where is Dickman's function.

     

On sums of a prime and four prime squares in short intervals

In this paper, we prove that each sufficiently large integer N ≠1（mod 3） can be written as N=p＋p1^2＋p2^2＋p3^2＋p4^2, with
｜p-N/5｜≤U,｜pj-√N/5｜≤U,j=1,2,3,4,
where U=N^2/20＋c and p,pj are primes.