Yes, there is a U.S. President with a connection to Utah:
Ezra Taft Benson
Position: Secretary of Agriculture under President Dwight D. Eisenhower (1953–1961).
Connection to Utah: Benson was a prominent member of The Church of Jesus Christ of Latter-day Saints (LDS Church) and was involved in politics in Utah, although he was never President of the United States. His association with Utah is notable, but he did not serve as President.
J. Reuben Clark
Position: He was the U.S. Ambassador to Mexico and a key figure in the LDS Church, but he was not a President of the United States.
As of now, there have been no U.S. Presidents who were directly from Utah. However, many notable political figures and leaders have had significant ties to the state.
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No, there has not been one yet. In 2012, the Republican nominee for president was a Mormon (Mitt Romney). His father, George Romney, tried to get the Republican nomination in 1964 and 1968, but was not successful.
No US president up to this time (June 2011) has been a member of the Church of Jesus Christ of Latter-day Saints (the "Mormon" church), although several Mormons have run for president over the years.
Ther are two members of the Church of Jesus Christ of Latter-day Saints (the "Mormon" church) running for president in the current (2012) presidential election: Mitt Romney and Jon Huntsman.
No president of the United States has been a member of the Church of Jesus Christ of Latter-day Saints (the "Mormon" church), although a handful of Mormons have been presidential candidates.
If you're asking whether any US Presidents were constitutionally ineligible to hold office, the answer is no.
There hasn't been any presidents from Rhode Island, yet.
No, none were.
As of September 2014, there are no living former presidents that live in Rhode Island. A couple of the former presidents live in Texas and New York.
Andrew Jackson