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Let P denote an almost-prime with at most r prime factors, counted according to multiplicity, and let E( N) denote the number of natural numbers not exceeding N that are congruent to 4 modulo 24 yet cannot be represented as the sum of three squares of primes and the square of one P. Then we have E( N)≪log N. This result constitutes an improvement upon that of D. I. Tolev, who obtained the same bound, but with P in place of P. [ABSTRACT FROM AUTHOR]

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 $$ N^{1 + |\mathcal{A}|/(9N)} $$. [ABSTRACT FROM AUTHOR]

We introduce a new class of lightlike submanifolds, namely, Screen Cauchy Riemann (SCR) lightlike submanifolds of indefinite Kaehler manifolds. Contrary to CR-lightlike submanifolds, we show that SCR-lightlike submanifolds include invariant (complex) and screen real subcases of lightlike submanifolds. We study some properties of proper totally umbilical SCR-lightlike submanifolds, their invariant (complex) and screen real subcases. [ABSTRACT FROM AUTHOR]