Brun’s constant and a division bug
The reciprocals of the primes diverge — slowly, as $\ln\ln x$, but they get there. Restrict to twin primes and the story flips. Brun (1919) proved by sieve that
$$B_2 \;=\; \Bigl(\frac13+\frac15\Bigr)+\Bigl(\frac15+\frac17\Bigr)+\Bigl(\frac1{11}+\frac1{13}\Bigr)+\cdots \;\approx\; 1.902160583\ldots$$converges.1 The sieve shows twins have density $O(x/\log^2 x)$ — thin enough to sum — while saying nothing about whether the sum has infinitely many terms. That remains open; the modern breakthrough is bounded gaps: infinitely many prime pairs differ by at most 246 (Zhang, then Maynard and Polymath).
1
The division bug
In 1994 Thomas Nicely was tabulating twin primes toward $B_2$ across two machines when the reciprocal sums began to disagree. He isolated the discrepancy to a single quotient on the new Pentium:
4195835 / 3145727
correct : 1.333820449136241002
Pentium : 1.333739068902037589Five entries were missing from the lookup table of the chip’s radix-4 SRT divider; roughly one in nine billion divisor pairs went wrong from the fourth significant digit on. Intel’s eventual no-questions recall cost about 475 million dollars — priced, ultimately, by a number theorist’s insistence on computing the same constant two ways.
That is the engineering content of $B_2$: independent recomputation is not paranoia. Nicely ran duplicate stacks and reconciled them; the discipline, not the arithmetic, caught the hardware. Any pipeline that computes a number once has no idea whether it computed it.