Which is the first integer that an IEEE 754 float is incapable of representing exactly?

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2mantissa bits + 1 + 1

The +1 in the exponent (mantissa bits + 1) is because, if the mantissa contains abcdef... the number it represents is actually 1.abcdef... × 2^e, providing an extra implicit bit of precision.

Therefore, the first integer that cannot be accurately represented and will be rounded is:
For float, 16,777,217 (224 + 1).
For double, 9,007,199,254,740,993 (253 + 1).

>>> 9007199254740993.0
9007199254740992

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