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Can my federal adjusted gross income differ by $1 between my federal and state returns?
No, you don't. Rounding errors happen, and if there's no change in the actual tax there's no reason to amend. If all the income was properly reported and the tax was properly calculated - no-one cares if it was rounded up or down on one of the lines. Note for the next time though: Not sure about New York, but Federal taxes are generally rounded to the nearest dollar on each line of the form. So don't calculate cents, just round to the nearest dollar, and be consistent on all of your tax forms. Technically, it is perfectly legal to report cents as well (and people used to when the forms were still filled by hand with pen), but all the automatic tax filing software rounds, so just do that.