Let denote the number of primes of the form for , then
(1)
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where is the logarithmic integral (Shanks 1960, pp. 321-332). Let denote the number of primes of the form for , then
(2)
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(Shanks 1961, 1962). Let denote the number of pairs of primes and for , then
(3)
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where
(4)
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(Shanks 1960, pp. 201-203). Finally, let denote the number of pairs of primes and for , then
(5)
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(Lal 1967), where is called Lal's constant. Shanks (1967) showed that .