Penner 1999，第34頁: Lemma B.2.2, The integer 0 is even and is not odd. Penner uses the mathematical symbol ∃, the existential quantifier, to state the proof: "To see that 0 is even, we must prove that ∃k (0 = 2k), and this follows from the equality 0 = 2 ⋅ 0."
David E. Joyce. 7. An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number.. Department of Mathematics and Computer Science Clark University. 1997 [2020-02-06]. （原始內容存檔於2020-02-03） （英語）. On Definition 6: The definition even number is clear: the number a is even if it is of the form b + b. The first few even numbers are 2, 4, 6, 8, 10. On Definition 7: The definition for odd number has two statements. The first can be taken as a definition of odd number, a number which is not divisible into two equal parts, that is to say not an even number. The first few odd numbers are 3, 5, 7, 9, 11. Euclid did not treat 1 as a number, but now 1 is also considered an odd number.
David E. Joyce. Definitions 6–7. Department of Mathematics and Computer Science Clark University. 1997 [2020-02-06]. （原始內容存檔於2020-02-03） （英語）. The other statement is not a definition for odd number, since one has already been given, but an unproved statement. It is easy to recognize that something has to be proved, since if we make the analogous definitions for another number, say 10, then analogous statement is false. Suppose we say a 「decade number」 is one divisible by 10, and and 「undecade number」 is one not divisible by 10. Then it is not the case that an undecade number differs by a unit from a decade number; the number 13, for instance, is not within 1 of a decade number. The unproved statement that a number differing from an even number by 1 is an odd number ought to be proved. That statement is used in proposition IX.22 and several propositions that follow it. It could be proved using, for instance, a principle that any decreasing sequence of numbers is finite.
Jonathan Hogeback. Is Zero an Even or an Odd Number?. Encyclopædia Britannica. [2020-02-06]. （原始內容存檔於2019-08-11） （英語）. So where exactly does 0 fall into these categories? Most people are confused by the number 0, unsure if it’s an integer to begin with and unaware of its placement as a number, because it technically signifies an empty set. Under the rules of parity, is zero even or odd?
Ball, Lewis & Thames (2008，第15頁) discuss this challenge for the elementary-grades teacher, who wants to give mathematical reasons for mathematical facts, but whose students neither use the same definition, nor would understand it if it were introduced.
Lichtenberg 1972，第535–536頁 "...numbers answer the question How many? for the set of objects ... zero is the number property of the empty set ... If the elements of each set are marked off in groups of two ... then the number of that set is an even number."
Lichtenberg 1972，第537頁; compare her Fig. 3. "If the even numbers are identified in some special way ... there is no reason at all to omit zero from the pattern."
Gowers 2002，第118頁 "The seemingly arbitrary exclusion of 1 from the definition of a prime … does not express some deep fact about numbers: it just happens to be a useful convention, adopted so there is only one way of factorizing any given number into primes." For a more detailed discussion, see Caldwell & Xiong (2012).
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