Julia uses native integer arithmetic by default and these operations are overflowing. You can use Big Integers:
julia> big(2)^1600
44462416477094044620016814065517364315819234512137839319418223093753683069769152238984782576173969417485953521141049383745107056455283979316385016701612810119562585078620415976730705698345087039035930761275083827265405596065418173652685035788898113991627042329246850314029877161622487411877779578892097029690461532001915311366862468942148892205997883828265721290296220249202674740669814705818564765009960300389641843321936008416473775144511929246788246559538970957296160626364645376