x + 1 would be a consecutive integer where x is an integer.
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Only if x is already an integer. For any integer x, 2x + 2 will be an even integer.
x+3 and x+4 would be consecutive integers.
Yes, plus 1 is consider as positive one (+1). All positive and negative numbers are considered as integer.
Every integer is either even (divisible by 2) or odd (not divisible by 2). Since an even number plus 1 is odd and an odd number plus one is even, because 1 does not divide 2. We know (n + 4) is odd. The next integer is (n + 4 + 1) = (n + 5), because an odd number plus 1 is even, (n + 5) is even. The integer after (n + 5) is (n + 6), since (n + 5) we know is even, (n + 6) must be odd. Since (n + 6) is the smallest integer that is greater than (n + 4) and is odd, so (n + 6) is the next odd integer.
This is (3x +2)2 so if x is an integer/3, this expression is the square of an integer so cannot be prime. Anything else, and it's not even an integer.