Here is Intel's summary of Moore's Law, which is not truly a law, but an observation in the form of a prediction: "Gordon Moore made his famous observation in 1965, just four years after the first planar integrated circuit was discovered. The press called it "Moore's Law" and the name has stuck. In his original paper, Moore observed an exponential growth in the number of transistors per integrated circuit and predicted that this trend would continue. Through Intel's relentless technology advances, Moore's Law, the doubling of transistors every couple of years, has been maintained, and still holds true today. Intel expects that it will continue at least through the end of this decade. The mission of Intel's technology development team is to continue to break down barriers to Moore's Law." Notice that it relates to transistors on a single chip, not processor power or speed per se. And notice that even Intel has realistic notions about its limits. As with any exponential progression, at some point it will simply become unsustainable, long before the point you ask about. 2.89*10^90 exceeds the number of atoms in the observable universe, if I'm not mistaken, let alone the number of atoms availbale for integrated curcuits. So the answer to your question is purely one of mathematics, not practical reality. Assuming we started with 2 (transistors/chip, hertz, flops, or any other measure), a lower number than any of these started with, we find that log2(2.89*10^90)=~300. So, it will take about 300 cycles of doubling to reach that number. At 1.5 years per cycle, that would be approximately 450 years. We don't have to know exactly where we started or when to figure what cycle we're on now. I don't know offhand the curent number of transistors per chip, but I do know the speed of supercomputers, and that will suffice to show how the determination of which cycle we're in would be made. And it will deal with Flops, the terms of your question. The fastest supercomputers now run on the order of 10^14 Flops, which puts us at about the 47th cycle (log2(10^14)), which would subtract maybe 70 years, and reach your target number in about 380 years. Again, Moore's Law did not address multi processor machines, only transistors on a chip, but anything you wish to double every 18 months will progress at the same rate. And for what it's worth, just to get an idea of the outrageous scope of the processing power we're talking about: The fastest current supercomputer is currently around 135 Teraflops (1.35*10^14). To keep the math simple, let's imagine 100 Teraflop (10^14) machines. If the 500 fastest computers in the world were 100 Teraflop each (they would average less than that in reality) and could be run in parallel, that would be 5*10^16 Flops. If we divide 10^14 into 10^90, we get 10^76, or the number of 100 Teraflop supercomputers running in parallel required to achieve that speed. Each of the 6 billion (10^9) people in the world would need to be capable of constructing, maintaining, and operating 10^67 of these supercomputers (10^76/10^9). And that's imagining that the earth could supply the natural resources, space, and power for them, which it would not come near. It's hard to imagine that the combined processing power in the entire world will ever reach 10^30 Teraflops, unless there is some totally unanticipated approach developed. I don't even think quantum computing could do it. Hmmm...maybe superstring computing?

We dont expect to. Moore's "law" was proven false, just a few weeks ago, when somebody got it into their head that "hey- chips can only get SO small", and thus the trend dies. However, I think it's a pretty narrow assessment, because frankly humans aren't smart enough to say with any certainty that they understand the way the universe works. If there is no limit, theoretically, to how big something can be (With the universe being infinate), then what makes us think there is a limit to how small something can be? At one time, humans thought 0 was the smallest number... but then we thought up decimals. At one time, humans thought atoms were as small as anything got, then we discovered sub-atomic particles. So until we discover a way to make atom-sized technology, Moore's law will slow to a halt. And I know you were looking for a math solution, but the true answer is that we dont know when we'll reach that many because Moore's law doesn't stand true.