[ October 13, 2003 ] [ Last update May 7, 2025 ]
Tautonymic numbers

I first read this name in one of Terry Trotter's webpages
where he uses the number 5757 to present a new number format.

The scientific name is commonly used in zoölogy for naming species
in which the epithet for the species is the same as that of the genus.
Some examples : the animal Gorilla gorilla ;
the filarial worm Loa loa ;
the greenfinch Chloris chloris ;
the bank swallow Riparia riparia .

Transferred to the world of numbers the analogy brings us to
numbers that can be split up into two equal non-palindromic halves
and each part with at least two different digits.
Otherwise we would have repdigits which are already wellknown.
Note that these tautonymic numbers are all 'base related' since
they are digit-dependent. Generalized tautonymic numbers are such
numbers where more than two identical parts are allowed.
Near tautonymic numbers (NTN) have equal length parts that differ
by one. NTN's are the only tautonyms that can be prime !

No better introduction to tautonymic numbers than pointing your attention
to a popular oldtimer from artists Zager and Evans with their song
 In the Year 2525  which is almost a tribute to this kind of numbers.
Other tautonyms that appear in the song are
3535, 4545, 6565 and 9595.
5555 is excluded as it is a repdigit
and 7510 together with 8510 don't serve our purpose either
since not tautonymic at all, despite the fact that it seems
that these should have been 7575 and 8585 in the logic of things
but were almost certainly changed for lyrical reasons.

This wonplate will collect tautonymic numbers that show
interesting properties or are curious in some way or other.
Of course we stick to the recreational mathematical world.

Allow me to start with the number of the song title 2525 itself.
This number can be expressed with the nine digits 1 through 9 as follows :

2525 = (9–2)⁴ + (7–8)⁵ + (6–1)³ → 924785613

6565 = (2–5)⁸ + (4–3)⁷ + (9–6)¹ → 258437961

9595 when raised to the fourth power is a number
whose eight distinct digits, in base 10,
occur with the same frequency of two.
9595⁴ = 8475784699200625
It is the largest solution in its class P₄(8.2).

Enjoy the following neat fact that
36363636364 * 63636363637 = 23140495868_23140495868
is yet another superb tautonym !

The 26^th decagonal (10-sided polygonal) number is 2626 !
(Sloane's A001107)

A triangular tautonymic number
5050 = the sum of the first 100 integers = (100²+100)/2

Interesting tautonyms with palindromic halves are allowed here:

The string formed by concatenating
the natural numbers from 7 to 7171 is probable prime.

A Kaprekar tautonymic number
7272 since
7272² = 5288_1984 and 5288 + 1984 = 7272 !

9696 * 2⁹⁶⁹⁶ + 9696 + 1 is prime.
Note that 9696 is not only tautonymic but also strobogrammatic.
(source: Prime Curios! 9696)

The decimal expansion of 499⁴⁹⁹ ends with the digits 499499.
(source: Prime Curios! 499)

From the Prime Curios! database :
For any 510510 consecutive integers,
5760 will have 17 as a lowest prime factor. [Schuler]
Note that 510510 equals 714 x 715, two successive numbers !
Pomerance found that the product of 715*715-1 (510510) equaled
the product of both their prime factors: 2*3*5*7*11*13*17. [McAlee]

The smallest (probable) prime of 35631 digits is situated at
a tautonymic displacement from the power of ten axis
namely 10³⁵⁶³⁰ + 239239

188188 is a tautonymic cuberoot of an exclusionary cube. But there are more...
188_188³ = 6664625956660672
418_418³ = 73253955072930632
488_488³ = 116563263575030272
4442_4442³ = 87673015533155613358888
8988_8988³ = 726305734421244276046272
9522_9522³ = 863604334140616884744648
Can you find larger examples ?
See at OEIS A029785 Ninedigits page 2

The smallest number with exactly 240 divisors is 720720.
Pari/GP code : for(i=1,oo, if(#divisors(i)==240,
print(i," has exactly ",#divisors(i)," divisors");break()))

Are you creative and adventurous enough to discover
more of these tautonymic numbers ?
I will accept submissions until the year 2525...
" If man is still alive.
If woman can survive, they may find. "

A calculation trick involving tautonyms.
By Terry Trotter (†) - [ October 24, 2003 ]

" Here is an old idea that may be worth considering.
It's a calculation trick that appeals to, and often amazes, people
who are not highly knowledgeable about number properties.
The conversation between the magician and the patsy usually goes like this :

The magician says : "Choose any 3-place number you wish and write it down."

Let's say, by way of example, the patsy choose 276.

Now the magician says : "Form a 6-place number by writing it beside itself."

For our example, this means 276276. A tautonym has been produced.

Magician : "Divide that result by 7." Result is 39468.

Magician : "Divide that result by 11." Result is 3588.

Magician : "Divide that result by 13, and report what you have."

Of course, the patsy now has the original number, 276, once again !
The trick is easily explained. Writing a 3-place number beside itself
is equivalent to multiplying it by 1001. But 1001 = 7 x 11 x 13.
So all the person is doing is dividing the large number by 7, 11 and 13,
in order to arrive at the original 3-digit choice. Simple, yes.
Profound, no. But it is interesting to many people for whom mathematics
is a great mystery. And proving it makes a good school math activity. "

Tautonymic numbers as products of two consecutives.
By Patrick De Geest - [ October 28, 2003 ]

Here is something that emerged while I was investigating
tautonymic numbers being the product of two consecutive integers.
Playing around with tautonymic numbers I came across a
well known 'prime' number P = 5882353. I say well known
since this number made its entrance already in the following
two puzzles pages.
http://www.primepuzzles.net/puzzles/puzz_104.htm
http://www.primepuzzles.net/puzzles/puzz_180.htm

P = 5882353 = 588² + 2353²

Now multiply this famous prime with another famous prime '13'
and we get 76470589 ! Call this number N and you will see
that N times (N+1) equals 58477510_58477510, a tautonym !!!

Moreover N = 76470589 when split into two halves and
given the same treatment as P = 5882353 gives
7647² + 0589² = 5882353 * 10 or ten times our initial P !!

ps. 7647 + 2353 = 10000 exactly
Is there even more to discover in this curious topic ?

ps. note also this infinite beast pattern
363 * 364 = 132_132
336633 * 336634 = 113322_113322
333666333 * 333666334 = 111333222_111333222
etc.

and the next one was spotted by Jean Claude Rosa
637 * 638 = 406_406
663367 * 663368 = 440056_440056
666333667 * 666333668 = 444000556_444000556
etc.

Tautonymic numbers of the form (N) * (N+1)

1. 132_132 = 363 * 364
2. 406_406 = 637 * 638
3. 510_510 = 714 * 715
4. 852_852 = 923 * 924
5. 7930_7930 = 8905 * 8906
6. 66942_66942 = 81818 * 81819
7. 113322_113322 = 336633 * 336634
8. 440056_440056 = 663367 * 663368
9. 5289256_5289256 = 7272727 * 7272728
10. 58477510_58477510 = 76470589 * 76470590
11. 111333222_111333222 = 333666333 * 333666334
12. 164378892_164378892 = 405436668 * 405436669
13. 183673470_183673470 = 428571429 * 428571430
14. 200444410_200444410 = 447710185 * 447710186
15. 206611570_206611570 = 454545454 * 454545455
16. 224376732_224376732 = 473684211 * 473684212
17. 277008310_277008310 = 526315789 * 526315790
18. 297520662_297520662 = 545454546 * 545454547
19. 305024040_305024040 = 552289815 * 552289816
20. 326530612_326530612 = 571428571 * 571428572
21. 353505556_353505556 = 594563332 * 594563333
22. 444000556_444000556 = 666333667 * 666333668
23. 479289942_479289942 = 692307693 * 692307694
24. 506156050_506156050 = 711446449 * 711446450
25. 581006406_581006406 = 762237762 * 762237763
26. 695569506_695569506 = 834008097 * 834008098
27. 739569252_739569252 = 859982123 * 859982124
28. 772853520_772853520 = 879120879 * 879120880
29. 814065240_814065240 = 902255640 * 902255641
30. 948726600_948726600 = 974025975 * 974025976
31. 962088780_962088780 = 980861244 * 980861245
( from here entries by JCR [ Nov 18, 2003 ] )
32. 6431722380_6431722380 = 8019801980 * 8019801981
33. 7157761512_7157761512 = 8460355496 * 8460355497
34. 9138301710_9138301710 = 9559446485 * 9559446486
35. 20050158756_20050158756 = 44777403627 * 44777403628
36. 21419301960_21419301960 = 46280991735 * 46280991736
37. 28857318490_28857318490 = 53719008265 * 53719008266
38. 30495351502_30495351502 = 55222596373 * 55222596374
39. 52431292270_52431292270 = 72409455370 * 72409455371
40. 54631379962_54631379962 = 73913043478 * 73913043479
41. 63689920156_63689920156 = 79805964787 * 79805964788
42. 66112433920_66112433920 = 81309552895 * 81309552896
43. 97015431556_97015431556 = 98496411892 * 98496411893

JCR noticed also this nice '0_n9_n0_n9_n0_n' pattern emerging from nr. 32

The method used by JCR :

T_T = T*(10^L+1) where L = number of digits of T.
One searches for N such that N*(N+1) = T*(10^L+1) thus
the largest primefactor of 10^L+1, P for example,
divides N, or divides N+1. So it is sufficient to try N
multiple of P or N+1 multiple of P. If by chance P is
large enough the results are obtained very quick but for
10^15+1, P = 9091 and so the progression isn't that rapid.

JCR found all solutions of T_T = N*(N+1) up to 38 digits for T_T.
There are a lot of solutions ! Too many to display all.
So for T_T's of 24 digits and more I will restrict myself
to list only the smallest and the largest.

Here is an explanation why N or N+1 cannot be prime :

We have : T_T = T*(10^L+1) = N*(N+1)
The number T_T is written with 2*L digits so N is written
obligatory with L digits hence :
N <= 10^L–1 (1)
Suppose now that N is prime then N+1 is divisible by
10^L+1 hence N+1 ⩾ 10^L+1 and so N ⩾ 10^L, which
contradics the inequality (1).
If N+1 is prime then N is divisible by 10^L+1,
hence N ⩾ 10^L+1, which contradicts again inequality (1).

Continuation of the table (smallest-largest only) :
(by JCR [ Dec 2-4, 2003 ] )

T_T with 24 digits : 5 solutions
44. 111133332222_111133332222 = 333366663333 * 333366663334
48. 928339282860_928339282860 = 963503649635 * 963503649636

T_T with 26 digits : 4 solutions
49. 2066115702480_2066115702480 = 4545454545455 * 4545454545456
52. 5769181397602_5769181397602 = 7595512752673 * 7595512752674

T_T with 28 digits : 11 solutions
53. 11126290222882_11126290222882 = 33356094230113 * 33356094230114
63. 96284719602360_96284719602360 = 98124777504135 * 98124777504136

T_T with 30 digits : 91 solutions
64. 101042810175330_101042810175330 =
317872317409569 * 317872317409570
154. 973101434819340_973101434819340 =
986459038591740 * 986459038591741

T_T with 32 digits : 17 solutions
155. 1147227785472790_1147227785472790 =
3387075117963565 * 3387075117963566
171. 9020805170520430_9020805170520430 =
9497791938403594 * 9497791938403595

T_T with 34 digits : 11 solutions
172. 12615748728591762_12615748728591762 =
35518655279432753 * 35518655279432754
182. 82736707922790756_82736707922790756 =
90959720713506347 * 90959720713506348

T_T with 36 digits : 5 solutions
183. 111111333333222222_111111333333222222 =
333333666666333333 * 333333666666333334
187. 788169507760196310_788169507760196310 =
887789112210887789 * 887789112210887790

T_T with 38 digits : 2 solutions
188. 1322314049586776860_1322314049586776860 =
3636363636363636364 * 3636363636363636365
189. 4049586776859504132_4049586776859504132 =
6363636363636363636 * 6363636363636363637

Tautonymic numbers as products of three consecutives.
K * (K+1) * (K+2) = T_T
By Jean Claude Rosa - [ November 6, 2003 ]

Thus far JCR only discovered one solution namely :
76 * 77 * 78 = 456_456

Near tautonymic numbers as products of two and more consecutives.
K * (K+1) = (T)_(T ± 1)
K * (K+1) * (K+2) = (T)_(T ± 1)
K * (K+1) = (T)_(T ± 1)_(T ± 2)
etc.

1. 78 * 79 = 61_62
2. 80919 * 80920 = 65479_65480
3. 91809 * 91810 = 84289_84290
4. 326510 * 326511 = 106609_106610
5. 475025 * 475026 = 225649_225650
6. 524975 * 524976 = 275599_275600
7. 673490 * 673491 = 453589_453590
8. 4323777 * 4323778 = 1869505_1869506
9. 4767132 * 4767133 = 2272555_2272556
10. 5232868 * 5232869 = 2738291_2738292
11. 5676223 * 5676224 = 3221951_3221952

781340266 * 781340265 = 610492_610491_610490

19 * 20 * 21 = 79_80

13 * 12 * 11 = 17_16

Can you find larger/other examples ?

Near tautonymic numbers of the form (T)_(T + 1) as SQUARES.

1. 183_184 = 428²
2. 328_329 = 573²
3. 528_529 = 727²
4. 715_716 = 846²
5. 6099_6100 = 7810²
6. 13224_13225 = 36365²
7. 40495_40496 = 63636²
8. 106755_106756 = 326734²
9. 453288_453289 = 673267²
10. 2066115_2066116 = 4545454²
11. 2975208_2975209 = 5454547²
12. 22145328_22145329 = 47058823²
13. 28027683_28027684 = 52941178²
14. 110213248_110213249 = 331983807²
15. 110667555_110667556 = 332667334²
16. 147928995_147928996 = 384615386²
17. 178838403_178838404 = 422892898²
18. 226123528_226123529 = 475524477²
19. 275074575_275074576 = 524475524²
20. 333052608_333052609 = 577107103²
21. 378698224_378698225 = 615384615²
22. 445332888_445332889 = 667332667²
23. 446245635_446245636 = 668016194²
24. 518348515_518348516 = 719964246²
25. 574930563_574930564 = 758241758²
26. 647238399_647238400 = 804511280²
27. 657515568_657515569 = 810873337²
28. 734693880_734693881 = 857142859²
29. 801777640_801777641 = 895420371²
30. 826446280_826446281 = 909090909²
31. 897506928_897506929 = 947368423²
32. 898802499_898802500 = 948051950²
33. 924910143_924910144 = 961722488²
( from here entries by JCR [ Nov 26, 2003 ] )
34. 1568473680_1568473681 = 3960396041²
35. 3647681599_3647681600 = 6039603960²
36. 4789624063_4789624064 = 6920710992²
37. 8315420899_8315420900 = 9118892970²
38. 13973312520_13973312521 = 37380894211²
39. 16311962328_16311962329 = 40388070427²
40. 20087347599_20087347600 = 44818910740²
41. 22873345935_22873345936 = 47826086956²
42. 27221172024_27221172025 = 52173913045²
43. 30449526120_30449526121 = 55181089261²
44. 35535821475_35535821476 = 59611929574²
45. 39211524099_39211524100 = 62619105790²
46. 80200635024_80200635025 = 89554807255²
47. 85677207840_85677207841 = 92561983471²
48. 94076078655_94076078656 = 96992823784²
49. 111066675555_111066675556 = 333266673334²
50. 444533328888_444533328889 = 666733326667²
51. 547194595608_547194595609 = 739726027397²
52. 859342532899_859342532900 = 927007299270²
***
A nice pattern emerges from this list (see nrs. 3, 9 & 22)
528_529 = 727²
453288_453289 = 673267²
445332888_445332889 = 667332667²

(4_n)5(3_n)2(8_n)8_(4_n)5(3_n)2(8_n)9 = (6_n)7(3_n)2(6_n)7²

Among the first 33333 values of n [ PDG, July 12, 2004 ]
only 0, 16, 17, 33, 2738, 3096 yield primes.
The last two are of course only probable primes of titanic
lengths 8217 and 9291 resp. ( using the formula 3*n+3 ).
Now that length 100000 is crossed the search has stopped.

The squareroot is an intertwining of
 727  with another palindrome 6_n3_n6_n !

Palindrome 909090909 (with nine digits btw!) squared
harbours another palindrome twice only composed of even digits.
909090909² = (82644628)0_(82644628)1
Note also that 82644628 * 11 + 1 = 909090909
***
Another nice pattern from JCR this time (see nrs. 0, 8, 15 & 49)
075_076 = 274²
106755_106756 = 326734²
110667555_110667556 = 332667334²
111066675555_111066675556 = 333266673334²

(1_n)0(6_n)7(5_n)5_(1_n)0(6_n)7(5_n)6 = (3_n)2(6_n)7(3_n)4²
Note that the squareroot is an intertwining of
 274  with palindrome 3_n6_n3_n !
xxx

 727  +  274  =  1001  a palindrome

 727  x  274  =  199_198  a tautonym of the form (T)_(T – 1)

Jean Claude Rosa found many new patterns :

3(9_n)6(0_n)3(9_n)6(0_n)4(0_m)1² with n odd and m = n-1
but alas, this number always seems to be divisible by 7.

Fascinating 9090909090911² ! Indeed, because JCR
discovered that this number is the first of yet another pattern !
Here it is :
If P = (90_n)911² with n = 11*k+5 then P^2 = T_(T+1)
PFGW was executed up to n = 50396 [ PDG, July 24, 2004 ]
Only one other probable prime (or PRP) with less
than 100000 digits popped up after the
first one with k = 0 and so n = 5 giving P = 9090909090911
whose square equals 8264462809920_8264462809921 .
Note the recurrent palindromic substrings in the expressions :
E.g. with 90909090909090909090909090909090911² =
82644628099173553719008264462809920_82644628099173553719008264462809921
This second probable prime is with k = 3219 and so n = 35414
and has 70831 digits in total, a personal record !

Two more patterns which, JCR hopes, might give better results :

P=(54_n)7 avec n = 11*k+3
One probable prime was found for n = 410 :
r(410,54)*10+7 is 3-PRP! (0.0542s+0.0171s)
A prime of only 821 digits.
I pushed the pattern up to n = 50064 [ PDG, July 31, 2004 ]
but alas no new candidate showed up below digitlength 100000 !

P=(72_n)7 with n = 11*k+1
Isn't this one particularly beautiful ?
Indeed a nice one and also known as a SUPP or
'Smoothly Undulating Palindromic Prime'.
Hans Rosenthal checked all probable primes candidates up to
40000 digits and I continued
up to 100001 [ PDG, September 13, 2004 ].
See reference table at undulat.htm
and go to section 727. None of the numbers in the table
are of the form 11*k+1. Pity :(

Near tautonymic numbers of the form (T)_(T – 1) as SQUARES.

1. 82_81 = 91²
2. 8242_8241 = 9079²
3. 9802_9801 = 9901²
4. 538277_538276 = 733674²
5. 998002_998001 = 999001²
6. 77837026_77837025 = 88225295²
7. 99980002_99980001 = 99990001²
( from here entries by JCR [ Nov 27, 2003 ] )
8. 7922547265_7922547264 = 8900869208²
9. 8643251345_8643251344 = 9296908812²
10. 9223797610_9223797609 = 9604060397²
11. 9999800002_9999800001 = 9999900001²
12. 106710893290_106710893289 = 326666333267²
13. 453378226757_453378226756 = 673333666734²
14. 491023832065_491023832064 = 700730927008²
15. 945958034530_945958034529 = 972603739727²
16. 999998000002_999998000001 = 999999000001²

discover with JCR this 'joli' pattern (see nrs. 1, 3, 5, 7, 11 & 16)

(9_n)8(0_n)2_(9_n)8(0_n)1 = (9_n+1)(0_n)1²
xxx

Add the square roots of index nrs. 12 & 13 to discover

326666333267 + 673333666734 = 1000000000001

their palindromic sum !

326666333267 * 673333666734 = 219955439977_219955439978

their product a near_tautonymic number of the form (T)_(T+1) !

The first pattern I investigated is (9_[n])(0_[n-1])(1)².
These numbers are prime only for the next four cases: n = 2, 4, 6 & 8.
None others below n = 50000 (PDG)! This number has some historical
trace as for instance the pattern for n = 2, 4, 6 & 8
appeared in Beiler's book (table 42 p. 85):
http://primes.utm.edu/curios/page.php?short=999999000001
and also
http://primes.utm.edu/curios/page.php?short=9901

Regarding p^2 = T_(T-1) here is a pattern (again by JCR!)
(he wrote: "a bit complicated but one does as he sees fit")
(3_k)2(6_n)(3_w)2(6_k)7² with n = 2*k+2, w = 2*k+1
Examples:
326666333267
332666666333332667
333266666666333333326667
and so on...
Here is the current state of affairs for the above
sibling-II pattern.
3-PRP! for the following values of k :
k = 9, 13, 40, 363 & 6305
none other below k = 16666 (PDG, length > 100000).
For k = 363 the total length (6*k+6) is 2184 digits.
For k = 6305 the numberlength is thus 37836 digits.
For Question Q2 of Puzzle 258 this is the best shot so far.
My first gigantic near-tautonymic number of the form p^2 = (T+1)_(T).