Trump, once the undisputed poll leader, is now running neck-and-neck with Carson in many opinion surveys.
Carson's ability to overcome his anger and an impoverished childhood to become a world-renowned neurosurgeon has been a central part of his personal story. He has written that on one occasion he nearly punched his mother and on another he attempted to stab a friend with a knife.
"I had what I only can label a pathological temper - a disease and this sickness controlled me, making me totally irrational," Carson has said.
"That's like, you know, I could say, they say you don't cure as an example, child molester. You don't cure these people. You don't cure the child molester."
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And during a rally Thursday night, Trump questioned Carson's description of racing to the bathroom in his house after one incident and beginning to pray for God's help in dealing with his temper.
"God heard my deep cries of anguish. A feeling of lightness flowed over me, and I knew a change of heart had taken place," Carson wrote in his memoir.
He added, "How stupid are the people of the country to believe this crap?"
A spokesman for Carson declined to comment on Trump's remarks.