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Universal Sequence: Definition

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What is a Universal Sequence?

In mathematics, a universal sequence can generate any sequence, within certain bounds. Simply put, it’s a sequence of integers with certain properties.

universal sequence

Could a universal sequence predictor create a universe?

A universal sequence predictor does exist (at least in theory), which can learn to predict any sequence in existence in the universe. However, the predictor requires too much computing time to be implemented in real life. [4] Because of this impracticality, definitions are normally strictly defined for specific purposes. The exact definition depends on what area of math you’re working with.

For example, a universal sequence for a group is a sequence of words (w1, w2, …) with the following property: The equations wn = sn (where n is a natural number) for any sequence in the group can be solved simultaneously from within the same group [1].

Another example, this time from number theory, which chiefly revolves around the ring of integers (ℤ). ℤ/nℤ, is obtained from universal sequence and has been used to solve an open problem concerning the existence of balanced Steinhaus triangles modulo a positive integer n [2].

Universal Sequence in Chemistry

In chemistry, a recent use of the term comes from Russian researchers Zahed Allahyari and Artem Oganov, who developed a new method for ordering elements of the periodic table. The universal sequence uses radii for atoms and electronegativity to make predictions about basic compounds [3].


[1] Hyde, J. et al. (2020). Sets of universal sequences for the symmetric group and analogous semigroups. Proc. Amer. Math. Soc. 148. Retrieved April 3, 2021 from:
[2] Jonathan Chappelon. A universal sequence of integers generating balanced Steinhaus figures modulo an odd number. Journal of Combinatorial Theory, Series A, Elsevier, 2011, 118 (1), pp.291-315. ⟨10.1016/j.jcta.2010.06.005⟩. ⟨hal-00409604v2⟩
[3] Z Allahyari and A R Oganov, (2020). Nonempirical Definition of the Mendeleev Numbers: Organizing the Chemical Space. J. Phys. Chem. C, 2020, 124, 23867 (DOI: 10.1021/acs.jpcc.0c07857)
[4] Hibbard, B. Adversarial Sequence Prediction. Retrieved April 4, 2020 from:

Stephanie Glen. "Universal Sequence: Definition" From Calculus for the rest of us!

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