Four Puzzles for the Price of One
Here is a math problem from the 1977 USSR Math Olympiad:
Let A be a 2n-digit number. We call this number special if it is a square and a concatenation of two n-digits squares. Also, the first n-digit square can’t start with zero; the second n-digit square can start with zero, but can’t be equal to zero.
- Find all two- and four-digit special numbers.
- Prove that there exists a 20-digit special number.
- Prove that not more than ten 100-digit special numbers exist.
- Prove that there exists a 30-digit special number.
Obviously, these questions are divided into two groups: show the existence and estimate the bound. Furthermore, this problem can be naturally divided into two other groups. Do you see them? The puzzle about special numbers makes a special day today — you get a four-in-one puzzle.
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Zak Seidov:
As to
* Prove that there exists a 20-digit special number.
Two smallest such numbers are
24 October 2008, 2:17 am10894620496601400001
24999000019999800001
Zak Seidov:
no more 20-digit special numbers!
24 October 2008, 2:46 pmPhilippe Fondanaiche:
The numbers 5*10^k – 1 and 10^(k+1) – 1 provide (4k+4)-digit special numbers for k = 0,1,2,3,…
Indeed 10^(2k+2)*(5*10^k – 1)^2 + (10^(k+1) – 1)^2 = (5*10^(2k+1) – 10^(k+1) + 1)^2
So there always exist special numbers whose length is a multiple of 4, particularly length = 20 and 100.
Examples 4^2 + 9^2 = 41^2, 49^2 + 99^2 = 4901^2, 499^2 + 999^2 = 499001^2, etc….,the above mentioned solution 49999^2 + 99999^2 = 4999900001^2
On the other hand, solutions can be found on the basis of the Pell equations of the form 10*x^2 + p^2 = y^2 with p = 1, 2, 3 or 1000*x^2 + p^2 = y^2 with p = 11,12,…
14 May 2009, 5:19 pmFor example we can obtain: x=18, p = 3 and y = 57 with the corresponding special number 324 900 = 570^2
or x = 228, p = 1, y = 721 with the special number 5 198 410 000 = 72100^2
etc…
Sergio Corazza:
30-digit special numbers are 136
Some example:
102399989626944262696022410000
159999983792100410462535015625
249999990486544090505845063936
390624985135225141415382912400
562499978594724203638151393856
810000003329001003420446808025
999999961946176362023380255744
100-digit special numbers are probably only 2:
(5*10^24-1)^2*10^50+(10^25-1)^2 and
2455090255207329259155412734890295397355737147032999097734690704797344551460000000000000000000000001
1232-digit special numbers are also probably 2:
8 March 2025, 1:07 pm(5*10^307-1)^2*10^50+(10^308-1)^2 and
15314511714329987236364062928550346144765044351100495806883291742940625107231215111029739674343046476181826587993345664853647893823275991719050434038354479818087330696845138769345444518101710654331254578018382690519046330801743978808471697586395239436034924250777659123132968891978938929025441782192813665576073860035496375994317318461491822740892009337665032826798253540391859106700944610318546384285059854480808199408749916267014340687914773384392074204464988426385012421993617008547879115831966338908029585461296115209311379900594770693198861737713856780848814564079908151632712171956408525564265001385425037296017826751999221640659168094788920174460382212425762060952275021586993759503419295957562224662055268761308286281180037616398450882196903200100711207652081612116101796368791612501124509711323853339576031521049209347303905664086115452245410776531849964506104305758446153601437124862335578775847440838737521512710200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001