Sunday 16 June 2024

Automorphic Numbers and some history notes

  As far back as the Babylonians, and maybe much before, someone looked at the symbol for five, and the symbol for 25 and noticed that both ended with the same symbol for five.  Even 5 x 25 ended in the symbol tor two x sixty and the same 5 symbol.  In Roman numerals V times V = XXV, and V times XXV  =  CXXV.  Even XXV times XXV , written as DCXXV repeats the original factor in its ending.  6^2 = 36 also appears easily in Roman numerals, VI x VI = XXXVI, and 76^2 as well.

It's hard to imagine that number crunchers like Diophantus didn't recognize some of these repetitive patterns, and if there is earlier mention, would one of those more knowledgeable historians give me a note.  

For at least 200 years, the one digit autonomic numbers, 0, 1, 5, and 6 have been called circular numbers. The earliest mention of the term I can find is An Introduction to the knowledge and variety of Numbers, John Smith (Schoolmaster of Norwich), 1809.  On page 104 he writes, "The numbers 5 and 6 are called circular numbers ; because , like the circle , terminating where it begins , these numbers , multiplied by themselves ever so often , always end in the same number : 5 by 5 make 25 , and that product  multiplied by 5 makes 125, So 6 by 6 makes 36, and 6 times this product make 216, etc."  This term and usage is still preserved in some recreational math books into the 21st Century. 

The expanse of this idea to numbers that repeated the last two, or three, or more digits took on the term automorphic numbers, (formed on oneself).  The first use of the term automorphic in mathematics, seems to be by Arthur Cayley in terms of functions, "... invariant with respect to a group of linear transformations of a certain kind leaving a certain function invariant.  This meaning was superseded by Felix Klein's choice of "a function f(x) is automorphic with respect to a group G, then f(Tₙ(x)) = f(x) for every element Tₙ of G."

The term automorphic number now means an n digit number that when squared has the number appear as the last n digits of the square.  thus 25^2 =625 and 376^2 = 141376.  Many more examples below.

The first instances of the use of autonomic numbers I can find is from the early 1940's.

Mathematical Recreations by Maurice Kraitchik, 1942

He explains the term in a footnote :

He explains that prime numbers will not work except for the trivial n=0 or 1, and illustrates numbers in base 6 and 10, He explains that prime numbers will not work as bases except for the trivial n=0 or 1, and he illustrates numbers in base 6 , And lists numbers that are automorphic in base 6, "which end in 4, 44, or 344 and 3, 13, 213, and so on are automorphic."  

Then he does the same with base 10.

He closes by providing "the last digits may be 3740081787109376 or 6259918212890625."  He nowhere mentions other composite numbers whose bases work, 12, 14, 15.

Early in the same year, a problem in The American Mathematical Monthly, Vol. 49, No. 2 (Feb., 1942), pp. 120-121 (2 pages)  used the term. 

Since this was new to me, I tested it with n=2, 90625^4.  It came out as, 67451572418212890625; the last 11 digits were 18212890625 which is itself automorphic , producing a larger number that ends in these same 11 digits.

The Previous reference is from a question in 1941 that show what are now called trimorphic numbers, by H. S. M. Coxeter, 

These numbers are the 13 six-digit trimorphic numbers,  109375, 109376, 218751, 281249, 390625, 499999, 500001, 609375, 718751, 781249, 890624, 890625, 999999. Two of these are the six digit autonomic numbers 109376 and 890625. Tri-morphic are n-digit numbers whose cubes preserve the original in their last n digits. All automorphic numbers are automatically tri-morphic, but there are others which are not automorphic.

The automorphic numbers in base 10 have two sets (ignoring the trivial 0, 1). The fives are mostly self describing, 5^2 =25, 25^2 = 625, 625^2 presents a glitch, since it produces 390625, there is not a four digit automorphic number in this series, but we extend to the next digit to get 90625 which when squared produces 8212890625, and we get the six digit automorph, 890625.
The six series is a little more complex, 6^2=36, but 36 is not automorphic, instead we use the tens compliment of 3, 7 as the lead digit, and 76 is the two digit automorph. 75^2 = 5776, but 776 is not a 3 digit automorph, but the tens compliment of 7 yields 376 which is the three digit automorph.

there is an interesting combination for the numbers in base ten. Notice that 6+5 = 10+1, and 25 + 76 = 100+1. 625 + 376 = 1000+1, and 90625 + 9376= 10000+1.

***There is a flaw in the notes below due to, I believe, a typo in the spreadsheet that I'm working to correct.
There are also a-automorphic numbers for which ax^2 preserves the n-digits of x, so 2-automorphic would be 2x^2, and contain numbers like 8, 88, 688, 4688, 54688...

The 3-automorphic numbers I have found end in 2, 5, and 7. (still exploring these). 2, 92, 792, ... ; 5, 75, 875,..; 7, 67, 667, 6667...;

I think any of these would be great explorations for even middle school students....or old retired guys like me, so enjoy!

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