## 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.

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 (w_{1}, w_{2}, …) with the following property: The equations w_{n} = s_{n} (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].

## References

[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: https://www.ams.org/journals/proc/2020-148-05/S0002-9939-2020-14881-2/

[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: https://www.ssec.wisc.edu/~billh/g/hibbard_agi.pdf

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