64th Annual STS (2004-2005)
Finalists
Samuel Mohun Bhagwat

MICHIGAN
Samuel Mohun Bhagwat, 16, of Livonia aims at classifying determinantal sequences for his Intel Science Talent Search mathematics project. A determinantal sequence is a sequence a_n of nonzero integers, one for each integer n, where the determinant a_na_n+3 - a_n+1a_n+2 takes on the same fixed nonzero value d for all choices of n. All such sequences have two useful invariants, ALPHA = (a₀ + a₂)/a₁ = (a₂ + a₄)/a₃ = ... and BETA = (a₁ + a₃)/a₂ = (a₃ + a₅)/a₄ = ... The sequence is reduced if no prime integer divides all the odd-indexed entries, and no prime integer divides all the even-indexed entries; the reduced sequences are the basic building blocks for all others. For reduced sequences, ALPHA and BETA are integers, and Sam classifies the reduced cases when 0 <= ALPHABETA <= 4. For ALPHA and BETA outside this range, he shows there are only finitely many reduced sequences for each choice of d. The son of Dr. Ashok Bhagwat and Helen Gay, Sam studied guitar and his father's native language, Marathi, the last two summers in India. At Winston Churchill High School he is captain of the varsity Quiz Bowl team, and his awards include a Grand Award at the Science and Engineering Fair of Metro Detroit. Sam plans to attend Harvard.