There is an equally simple reason that neither of these predictions is likely to be realized. It simply takes more than 100 years for a large, less economically developed country to catch up with the world leader in per capita income. One need look only at the history of the United States, which had a much higher growth rate than Britain in the 19th century, yet did not catch up until World War I. Or consider Japan and the United States. Some 150 years after Japan started to modernize during the Meiji restoration, the country’s per capita G.D.P. is still only 80 percent of that of the United States in terms of purchasing power parity — although, in nominal terms, it has caught up.

The United States is not standing still. In fact, its per capita income grew faster than nearly all other big countries from 1990 to 2007. Europe’s per capita income fell from 85 percent of that of the United States in 1990 to 66 percent in 2007, according to International Monetary Fund statistics.

So let’s say that the inflation-adjusted growth rate for China is 4 percent a year. This is optimistic, because China will certainly have some bad years in the next century. Every country does — remember the Great Depression in the United States. A 4 percent rate is faster than any big country has ever grown for 100 years. But assume that China can do it. Assume, too, that America grows at the 3 percent rate it has averaged for the last 15 years.

Now project the two growth rates forward: the inflation-adjusted per-capita G.D.P. of China would be less than $40,000 in 2100, versus almost $650,000 in the United States. That’s because China starts at $1,000 per capita and the United States at $43,000. If, in 2100, China has four times as many people as the United States, as it does now, China would still not have a total G.D.P. equal to America’s.

But it is unlikely to have four times as many people. It is always a mistake to project population growth rates for a century, but let’s do it anyway: With a one-child policy and a sex ratio that favors boys (many men won’t find wives) — China should experience a decline in population in the 21st century. Yet let’s assume for a moment that China’s population remains constant, at 1.3 billion. If immigration to the United States continued at the current rate, America’s population would rise. If the population grew at 1 percent a year, as it has recently, it would more than double by 2100, reducing the enormous population gap between the two countries. Are these projections likely to be realized? Who knows?

What is clear is that China is unlikely to surpass the United States in G.D.P. in absolute or relative terms anytime soon.

There may be a Chinese century, but it will be the 22nd century — not the 21st.