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Conference in memory of A. A. Karatsuba on number theory and applications
January 31, 2014 12:00–12:25, Moscow, Steklov Mathematical Institute, Conference-Hall
 


On some diophantine inequalities involving primes

S. A. Gritsenko
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S. A. Gritsenko


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Abstract: We present the following result obtained by the speaker:
Theorem 1. Suppose that $H>\sqrt{N}\exp(-\ln^{0.1}N)$. Then the inequality
$$ |p_1^2+p_2^2-N|\leqslant H $$
is solvable in primes $p_1$ and $p_2$.
Theorem 2. Let $H>N^{ 49/144}\exp(\ln^{0.8}N)$. Then the inequality
$$ |p_1^2+p_2^2+p_3^2-N|\leqslant H $$
is solvable in primes $p_1$, $p_2$ and $p_3$.
Theorem 3. Suppose that $H>N^{ 7/72}\exp(\ln^{0.8}N)$. Then the inequality
$$ |p_1+p_2-N|\leqslant H $$
is solvable in primes $p_1$ and $p_2$.
The proof of Theorem 1 would be also presented.

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