Automorphe number

In Mathematical, a number automorphe is a Integer whose Carré ends in the same figure or the same figures as that or those of the number itself. For example, 5² = 2 5 , 76² = 57 76 , and 890625² = 793212 890625 .

Being given a automorphe number to K figures, a automorphe number to 2 K figures can be obtained by

$n\text{'}\; \backslash \; equiv\; 3\; \backslash \; cdot\; n^2\; -\; 2\; \backslash \; cdot\; n^3\; \backslash \; (10^\; \{2k\})$

There exist two automorphes numbers of K figures. One of them checks the conditions

$n\; \backslash \; equiv\; 0\; \backslash \; (2^\; \{K\}),\; \backslash \; qquad\; N\; \backslash \; equiv\; 1\; \backslash \; (5^\; \{K\})$

and the other checks

$n\; \backslash \; equiv\; 1\; \backslash \; (2^\; \{K\}),\; \backslash \; qquad\; N\; \backslash \; equiv\; 0\; \backslash \; (5^\; \{K\})$.

The sum of the two automorphes numbers is worth 10^k + 1.

The following continuation makes it possible to find a automorphe number with K figures, where K ≥ 1000.

12781254001336900860348890843640238757659368219796 \ 26181917833520492704199324875237825867148278905344 \ 89744014261231703569954841949944461060814620725403 \ 65599982715883560350493277955407419618492809520937 \ 53026852390937562839148571612367351970609224242398 \ 77700757495578727155976741345899753769551586271888 \ 79415163075696688163521550488982717043785080284340 \ 84412644126821848514157729916034497017892335796684 \ 99144738956600193254582767800061832985442623282725 \ 75561107331606970158649842222912554857298793371478 \ 66323172405515756102352543994999345608083801190741 \ 53006005605574481870969278509977591805007541642852 \ 77081620113502468060581632761716767652609375280568 \ 44214486193960499834472806721906670417240094234466 \ 19781242669078753594461669850806463613716638404902 \ 92193418819095816595244778618461409128782984384317 \ 03248173428886572737663146519104988029447960814673 \ 76050395719689371467180137561905546299681476426390 \ 39530073191081698029385098900621665095808638110005 \ 57423423230896109004106619977392256259918212890625

It is enough to take the K last figures. The oblique bar reverses means that the writing of the number continues with the following line. The other automorphe number is obtained by withdrawing the number of 10^k + 1.

Count of the automorphes numbers

Generalization

That is to say a base B , in the preceding case base 10.

Then it is a question of determining, for a N given, the , such as

$a^2\; \backslash \; equiv\; has\; \backslash \; MOD\; b^n$

One can thus say that the automorphes numbers correspond to the fixed points of the square application, $f\; (X)\; =\; x^2\; \backslash ,$, other that 0 and 1, in the powers of the base.

One can prove that for a base B K > 1 produces distinct prime numbers, then there is K solutions by the Théorème of the Chinese remainders.

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