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Lal's Constant


Let P(N) denote the number of primes of the form n^2+1 for 1<=n<=N, then

 P(N)∼0.68641li(N),
(1)

where li(N) is the logarithmic integral (Shanks 1960, pp. 321-332). Let Q(N) denote the number of primes of the form n^4+1 for 1<=n<=N, then

 Q(N)∼1/4s_1li(N)=0.66974li(N)
(2)

(Shanks 1961, 1962). Let R(N) denote the number of pairs of primes (n-1)^2+1 and (n+1)^2+1 for n<=N-1, then

 R(N)∼0.48762li_2(N),
(3)

where

 li_2(N)=int_2^N(dn)/((lnn)^2)
(4)

(Shanks 1960, pp. 201-203). Finally, let S(N) denote the number of pairs of primes (n-1)^4+1 and (n+1)^4+1 for n<=N-1, then

 S(N)∼lambdali_2(N)
(5)

(Lal 1967), where lambda is called Lal's constant. Shanks (1967) showed that lambda approx 0.79220.


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References

Lal, M. "Primes of the Form n^4+1." Math. Comput. 21, 245-247, 1967.Shanks, D. "On the Conjecture of Hardy and Littlewood Concerning the Number of Primes of the Form n^2+a." Math. Comput. 14, 321-332, 1960.Shanks, D. "On Numbers of the Form n^4+1." Math. Comput. 15, 186-189, 1961.Shanks, D. Corrigendum to "On the Conjecture of Hardy and Littlewood Concerning the Number of Primes of the Form n^2+a." Math. Comput. 16, 513, 1962.Shanks, D. "Lal's Constant and Generalization." Math. Comput. 21, 705-707, 1967.

Referenced on Wolfram|Alpha

Lal's Constant

Cite this as:

Weisstein, Eric W. "Lal's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LalsConstant.html

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