GREENVILLE, Ill. — A lifer with a pen sat in the 65-square-foot cell he shares. A calendar taunted from a bulletin board. He began to write.

Dear President Obama.

He acknowledged his criminal past. He expressed remorse. And he pleaded for a second chance, now that he had served 18 years of the worst sentence short of execution: life without parole, for a nonviolent first offense.

Mr. President, he wrote, “you are my final hope.”

Sincerely, Jesse Webster.

Eleven hundred men reside in medium security at this remote Greenville federal prison in Southern Illinois. Most come and go, sentences served. Others stay, their legal appeals exhausted, their only hope to take up a pen and enter the long-shot lottery of executive clemency with a salutation that begins: Dear President Obama.

The prisoners are men like Mr. Webster, 46, a former cocaine dealer from the South Side of Chicago. And his old cellmate, Reynolds Wintersmith Jr., 39, a former teenage crack-cocaine dealer from Rockford, Ill., who has spent half his life in prison.