The Famous Bear Stearns Study, now 5 years old. Conclusion: illegals > 20 million. They even found that illegals held 15 million jobs in the U.S., and even conservative assumptions about family size would add 5 - 10 million to that -- and that was five years ago!
Wikipedia review including Bear Stearns and Pew. Here they quantify the Bear Stearns number at 20-22 million. Still low.
The population activist Fred Elbel arrives at a number well over 20 million, and that was 4 years ago. His methods and numbers are sound, and the Border Patrol numbers agree.
The number you recite uses the "Residual Method", which has serious flaws: Reasonable critique here.
There are many other methods, the Bear Stearns study being the more well known. But simple arithmetic and reasonable assumptions should tell you the story: it's not just half a million a year net. More like 3 to 6 times that: 1.7 million apprehensions a year. Assume - conservatively - that they get 1 in 3. That means a residual of 5.1 million. OK! Too high for ya? Say 50% apprehension rate. Then it's 2.55 million. That's 25 million in 10 years, starting from ZERO.
Lots of ways to dice those numbers. But the reality is that the Pew numbers are nonsense, and the real numbers are far higher. Many studies have concluded just that.
It's north of 22-25 million. Easy.
Bear Stearns is not even around anymore, and they are an odd source for the illegal population in America, they are not a credible source. I figured that you would not accept any valid sources, from Pew to the Federal agencies, and I was right.