Equinumerous

Let $A$ and $B$ be two classes of Positive integers. Let $A(n)$ be the number of integers in $A$ which are less than or equal to $n$, and let $B(n)$ be the number of integers in $B$ which are less than or equal to $n$. Then if

$$A(n)\sim B(n),$$

$A$ and $B$ are said to be equinumerous.

The four classes of Primes $8k+1$, $8k+3$, $8k+5$, $8k+7$ are equinumerous. Similarly, since $8k+1$ and $8k+5$ are both of the form $4k+1$, and $8k+3$ and $8k+7$ are both of the form $4k+3$, $4k+1$ and $4k+3$ are also equinumerous.

See also Bertrand's Postulate, Choquet Theory, Prime Counting Function

References

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 21-22 and 31-32, 1993.