Lal's Constant

Let $P(N)$ denote the number of Primes of the form $n^2+1$ for $1\leq n\leq N$, then

$$P(N)\sim 0.68641\mathop{\rm li}\nolimits (N),$$ (1)

where $\mathop{\rm li}\nolimits (N)$ is the Logarithmic Integral (Shanks 1960, pp. 321-332). Let $Q(N)$ denote the number of Primes of the form $n^4+1$ for $1\leq n\leq N$, then

$$Q(N)\sim {\textstyle{1\over 4}}s_1\mathop{\rm li}\nolimits (N)=0.66974\mathop{\rm li}\nolimits (N)$$ (2)

(Shanks 1961, 1962). Let $R(N)$ denote the number of pairs of Primes $(n-1)^2+1$ and $(n+1)^2+1$ for $n\leq N-1$, then

$$R(N)\sim 0.48762\mathop{\rm li}\nolimits _2(N),$$ (3)

where

$$\mathop{\rm li}\nolimits _2(N)\equiv \int_2^N {dn\over(\ln n)^2}$$ (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 $n\leq N-1$, then

$$S(N)\sim \lambda\mathop{\rm li}\nolimits _2(N)$$ (5)

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

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.