Squares In Arithmetic Progression Research Articles

Words avoiding repetitions in arithmetic progressions

Carpi constructed an infinite word over a 4-letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference. We show a connection between Carpi’s construction and the paperfolding words. We extend Carpi’s result by constructing uncountably many words that avoid squares in arithmetic progressions of odd difference. We also construct infinite words avoiding overlaps and infinite words avoiding all sufficiently large squares in arithmetic progressions of odd difference. We use these words to construct labelings of the 2-dimensional integer lattice such that any line through the lattice encounters a squarefree (resp. overlapfree) sequence of labels.

Almost squares in arithmetic progression (III)

It is proved that a product of four or more terms of positive integers in arithmetic progression with common difference a prime power is never a square. More general results are given which completely solve (1.1) with gcd(n, d)=1, k≥3 and 1<d≤104.

Almost squares in arithmetic progression (II)

Almost squares in arithmetic progression (II)

Almost Squares in Arithmetic Progression

It is proved that a product of four or more terms of positive integers in arithmetic progression with common difference a prime power is never a square. More general results are given which completely solve (1.1) with gcd(n, d)=1, k≥3 and 1&lt;d≤104.

Squares in arithmetic progressions

Squares in arithmetic progressions

On Squares in Arithmetic Progression

MERTON Taylor Goodrich, in an article in the Mathematics Magazine (1966), developed a method to obtain square integers in arithmetic progression.

On Squares in Arithmetic Progression

(1966). On Squares in Arithmetic Progression. Mathematics Magazine: Vol. 39, No. 2, pp. 87-88.

On Squares in Arithmetic Progression

On Squares in Arithmetic Progression