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In what is known in the United States as traditional mathematics, a specific process is taught to students at the end of the 1st year (or during the 2nd year) for use with multi-digit whole numbers, and is extended in either the fourth or fifth grade to include decimal representations of fractional numbers.
the number N+1 is 'cold' if only 'hot' numbers can be reached by subtracting a positive square; the number N+1 is 'hot' if at least one 'cold' number can be reached by subtracting a positive square; Using this algorithm, a list of cold numbers is easily derived: 0, 2, 5, 7, 10, 12, 15, 17, 20, 22, 34, 39, 44, … (sequence A030193 in the OEIS)
Numbers of the form 31·16 n always require 16 fourth powers. 68 578 904 422 is the last known number that requires 9 fifth powers (Integer sequence S001057, Tony D. Noe, Jul 04 2017), 617 597 724 is the last number less than 1.3 × 10 9 that requires 10 fifth powers, and 51 033 617 is the last number less than 1.3 × 10 9 that requires 11.
For example, the ascending interval from C to the next F is a perfect fourth, because the note F is the fifth semitone above C, and there are four staff positions between C and F. Diminished and augmented fourths span the same number of staff positions, but consist of a different number of semitones (four and six, respectively).
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