On A Conjecture By Russo

Charles Ashbacher

Charles Ashbacher Technologies
Hiawatha, IA USA
e-mail 71603.522@compuserve.com


The Smarandache Square-Partial-Digital Subsequence(SSPDS) is the sequence of square integers which can be partitioned so that each element of the partition is a perfect square[1][2][3]. For example, 3249 is in SSPDS since 3249 can be partitioned into 324 = 182 and 9 = 32.

The first terms of the sequence are:

49, 144, 169, 361, 441, 1225, 1369, 1444, 1681, 1936, 3249, 4225, 4900, 11449, 12544, 14641, . . .

where the square roots are

7, 12, 13, 19, 21, 35, 37, 38, 41, 44, 57, 65, 70, 107, 112, 121, . . .

this sequence is assigned the identification code A048653[4].

L. Widmer examined this sequence and posed the following question[2]:

     Is there a sequence of three or more consecutive integers whose squares are in SPDS?

For the purposes of this examination, we will assume that 0 is not a perfect square. For example, 90 will not be considered a number that can be partitioned into two perfect squares. Furthermore, elements of the partition are not allowed to have leading zeros. For example, 101 cannot be partitioned into perfect squares.

Russo[5] considered this question and concluded that the only additional solution to the Widmer question up to 3.3E+9 was

n n2Partition
     12225           149450625            1,4,9,4,50625     
     12226           149475076            1,4,9,4,75076     
     12227           149499529            1,4,9,4,9,9,529     

and made the following conjecture:

      There are no four consecutive integers whose squares are in SSPDS.

The purpose of this short paper is to present several additional solutions to the Widmer question as well as a counterexample to the Russo conjecture.

A computer program was written in the language Delphi Ver. 4 and run for all numbers n, where n <= 100,000,000 and the following ten additional solutions were found

n n2Partition
     376779           141962414841            1,4,1,9,6241,4,841     
     376780           141963168400            1,4,196,3168400     
     376781           141963921961            1,4,196392196,1     

n n2Partition
     974379           949414435641            9,4,9,4,1,4,4356,4,1     
     974380           949416384400            9,4,9,4,16,384400     
     974381           949418333161            9,4,9,4,1833316,1     

n n2Partition
     999055           998110893025            9,9,81,1089,3025     
     999056           998112891136            9,9,81,1,289,1,1,36     
     999057           998114889249            9,9,81,1,4,889249     

n n2Partition
     999056           998112891136            9,9,81,1,289,1,1,36     
     999057           998114889249            9,9,81,1,4,889249     
     999058           998116887364            9,9,81,16,887364     

n n2Partition
     2000341           4001364116281            400,1,36,4,116281     
     2000342           4001368116964            400,1,36,81,16,9,64     
     2000343           4001372117649            400,1,3721,1764,9     

n n2Partition
     2063955           4259910242025            4,25,9,9,1024,2025     
     2063956           4259914369936            4,25,9,9,1,4,36,9,9,36     
     2063957           4259918497849            4,25,9,9,1849,784,9     

n n2Partition
     2083941           4342810091481            43428100,9,1,4,81     
     2083942           4342814259364            434281,4,25,9,36,4     
     2083943           4342818427249            434281,842724,9     

n n2Partition
     4700204           22091917641616            2209,1,9,1764,16,16     
     4700205           22091927042025            2209,1,9,2704,2025     
     4700206           22091936442436            2209,1,9,36,4,42436     

n n2Partition
     5500374           30254114139876            3025,4,1,1,4,139876     
     5500375           30254125140625            3025,4,1,25,140625     
     5500376           30254136141376            3025,4,1,36,141376     

n n2Partition
     80001024           6400163841048576            6400,16384,1048576     
     80001025           6400164001050625            6400,1,6400,1050625     
     80001026           6400164161052676            6400,1,64,16,1052676     

n n2Partition
     92000649           8464119416421201            8464,1,1,9,4,16,421201     
     92000650           8464119600422500            8464,1,19600,4,22500     
     92000651           8464119784423801            8464,1,1,9,784,423801     

Pay particular attention to the four consecutive numbers 999055, 999056, 999057 and 999058. These four numbers are a counterexample to the conjecture by Russo.

Given the frequency of three consecutive integers whose squares are in SSPDS, the following conjecture is made:

       There are an infinite number of three consecutive integer sequences whose squares are in SSPDS.

In terms of larger sequences, the following conjecture also appears to be a safe one:

      There is an upper limit to the length of consecutive integer sequences whose squares are in SSPDS.

We close with an unsolved question:

      What is the length of the largest sequence of consecutive integers whose squares are in SSPDS?

References

[1] Sylvester Smith, "A Set of Conjectures on Smarandache Sequences", Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.

[2] L.Widmer, "Construction of Elements of the Smarandache Square-Partial-Digital Sequence", Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 145-146.

[3] C. Dumitrescu and V. Seleacu, Some notions and questions in Number Theory, Erhus University Press, Glendale, Arizona, 1994.

[4] N. Sloane, "On-line Encyclopedia of Integer Sequences", http://www.research.att.com/~njas/sequences.

[5] F. Russo, "On An Unsolved Question About the Smarandache Square-Partial-Digital Subsequence" http://www.gallup.unm.edu/~smarandache/russo1.htm.