I agree with Noam (great pictures!) Some of the links from comments are quite informative. Here are two specific cases

1. D. N. Lehmer first used a device made of bicycle chains and rods and later devices made of gears with holes in them. This article relates in rather breathless prose that one of the gear mechanisms (combined with theory) proved in a few minutes that

the great unconquered nineteen digit number $3,011,347,479,614,249,131$ , known to be a factor of $2^{95} + 1$ and suspected to be prime, is in fact prime.

2. A few years back (maybe 2000?) there was great excitement over rumors that Adi Shamir had a breakthrough which would speed up (the sieving step of) the number theory sieve by "several orders of magnitude". The then record factoring of an RSA key was for a 465 bit integer and the breakthrough was rumored to make 512 bit keys "very vulnerable." When the details came out of the punnily named TWINKLE device it turned out to be an electro optical device using LEDs and filters. It is agreed to be quite clever, it has never been built. Enhanced (theoretical) versions might threaten 768 bit keys in under 9 months (That estimate was in 2000, for an organization willing to invest in 80,000 pentium 2 PCS and 5000 TWINKLE devices ). 1024 bit keys would probably have been well beyond that. I think that the state of the art in unbuilt (or so they say...) special purpose devices is no longer optical.

I agree with Noam (great pictures!) Some of the links from comments are quite informative. Here are two specific cases

1. D. N. Lehmer first used a device made of bicycle chains and rods and later devices made of gears with holes in them. This article relates in rather breathless prose that one of the gear mechanisms (combined with theory) proved in a few minutes that

   the great unconquered nineteen digit number $3,011,347,479,614,249,131$ , known to be a factor of $2^{95} + 1$ and suspected to be prime, is in fact prime.

2. A few years back (maybe 2000?) there was great excitement over rumors that Adi Shamir had a breakthrough which would speed up (the sieving step of) the number theory sieve by "several orders of magnitude". The then record factoring of an RSA key was for a 465 bit integer and the breakthrough was rumored to make 512 bit keys "very vulnerable." When the details came out of the punnily named TWINKLE device it turned out to be an electro optical device using LEDs and filters. It is agreed to be quite clever, it has never been built. Enhanced (theoretical) versions might threaten 768 bit keys in under 9 months (That estimate was in 2000, for an organization willing to invest in 80,000 pentium 2 PCS and 5000 TWINKLE devices ). 1024 bit keys would probably have been well beyond that. I think that the state of the art in unbuilt (or so they say...) special purpose devices is no longer optical.