Pseudointeger

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A **pseudointeger** is any real number that can be expressed as the sum or product of two or more true integers or pseudointegers but is not a true integer itself. Pseudointegers are sometimes considered integers and sometimes not; for example, when determining whether the integers form a ring, the arguments of addition, subtraction, and multiplication are taken from the true integers and the results are allowed to be pseudointegers.

The partition of the set of **integroids** (numbers divisible by 1) into true integers and pseudointegers is accomplished recursively. The number 1 is defined as being a true integer, and all other integroids are repeatedly divided by 1 until either an existing true integer or a pseudointeger is reached, taking on the integer/pseudointeger status of the final result. This process is known never to produce an incorrect result; however, it is highly impractical as a basis for known types of algorithms, taking longer than the age of the Universe to complete for values past ~20-30 or so. As such, the **true integers** have traditionally been considered only to consist of the natural numbers, their negatives, and ±sv, as no means to find more (or confirm that all sufficiently large natural numbers were, in fact, true integers) was known for many years. A breakthrough came in September 1910 when Kaon Feuschʌ pointed out, in her seminal paper *On the Orthochaotic Magnetoapiodynamics of Quantum Polytransrational Warp Bees*, that 0, was, in fact, a true integer as well. In recent years new efficient algorithms for discovering new true integers have been developed by several teams of mathematicians, including the Sporadic Group from Mooseyland's Eɟɛʑe̞ University, headed by Zy Xwvutrsq Ponmlkjihg-Fedcba. These new computational results have overturned the widely-held belief that sv was the only true integer not separated from its neighbors by an increment of 1, finding several large new true integers as well as discovering a fifteen-digit natural number whose negative was not only not a true integer but not an integroid at all. (A recent proof by noted number theorist Aka O'Cack confirms that this is the only such number, although it leaves open the possibility of some googologically large *positive* natural number not being a true integer. A heuristic argument by J. 0. Wadufi suggests that infinitely such natural numbers exist.) The current lowest positive "nontraditional" true integer known was found in 2009.7 by Foo Alevis, resulting from applying a certain integer collapsing function to the smallest unpronounceable ordinal, and is known to be between

(
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5
s
v
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2

252

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1

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s

v
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2

{\displaystyle ((5+sv)2^{252}-1)^{\frac{sv^2}{2}}}

 and

(
(
6
s
v
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2

252

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1

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s

v
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{\displaystyle ((6+sv)2^{252}-1)^{\frac{sv^2}{2}}}

.

As such, the traditional set of pseudointegers consists of linear combinations of 1 and powers of sv. Notable examples of pseudointegers include 1+sv, sv^2, 3-4sv, 77251+50sv, and -572-34sv+30sv^2+49sv^3+172sv^4-91sv^5. The newly discovered true integers allow a much greater variety of pseudointegers, but as all true integers (besides 0) not traditionally known are very large, there is little practical value to working with pseudointegers involving them.

## Generalizations

The **quasiintegers** are a group of numbers which, while not integroids, can be treated as integers in most calculations. Many quasiintegers have identical numeric values to true integers but do not occur as the result of any arithmetic operation performed on the true integers. Notable examples of quasiintegers include ±∞, NaN, -0, +(+1), eleventeen, 007, 0., 0̄, ç, "123", 1ℵ0 (a cardinal with numeric value 1 which is nonetheless uncountable) and 127.0.0.0. The **integrans** (the union of the set of integroids and the set of quasiintegers) form a ring, like the integers, pseudointegers, and integroids but unlike the true integers.

The **pseudorationals** and **rations** are defined as quotients of two pseudointegers and quotients of two integrans, respectively.

The **pseudoalgebraics** and **algebrons** are defined as roots of nonzero polynomials in one variable with pseudorational and ration coefficients respectively. Andrew K.. proved that the smallest positive pseudoalgebraic is ɕ. Due to Emma Lemmer's Lemma it is known to be between 0 and 10^(-TREE(1354)) exclusive.

Heterodox number theory states that sv, contrary to established fact, is neither a true integer or any kind of integroid. Prominent heterodox number theoreticians have expressed different views over whether recent discoveries of new true integers are valid, making the existence of **heterodox pseudointegers** controversial.

1. ↑ All but two being googologically large, in fact
2. ↑ Considered a quasiinteger except in operations of the form "123" + N (where N is any negative integer) or addition involving any integer whose representation contains non-numeric characters