On A Conjecture By Russo
Charles Ashbacher Technologies
Hiawatha, IA USA
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 . 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.
L. Widmer examined this sequence and posed the following question:
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 considered this question and concluded that the only additional solution to the Widmer question up to 3.3E+9 was
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
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?
 Sylvester Smith, "A Set of Conjectures on Smarandache Sequences", Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, 101-107.
 L.Widmer, "Construction of Elements of the Smarandache Square-Partial-Digital Sequence", Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 145-146.
 C. Dumitrescu and V. Seleacu, Some notions and questions in Number Theory, Erhus University Press, Glendale, Arizona, 1994.
 N. Sloane, "On-line Encyclopedia of Integer Sequences", http://www.research.att.com/~njas/sequences.
 F. Russo, "On An Unsolved Question About the Smarandache Square-Partial-Digital Subsequence" http://fs.unm.edu/russo1.htm.