Page 247 - Determinants and Their Applications in Mathematical Physics

232 5. Further Determinant Theory

5.8.5 Hessenberg Determinants with Prime Elements
Let the sequence of prime numbers be denoted by {p n } and define a
Hessenberg determinant H n (Section 4.6) as follows:

p 1 p 2 p 3 p 4 ···
1 p 1 p 2 p 3 ···

1 p 1 p 2 ··· .

H n =
1

p 1 ···

··· ···
n
This determinant satisfies the recurrence relation
n−1

H n = (−1) p r+1 H n−1−r , H 0 =1.
r
r=0
A short list of primes and their associated Hessenberg numbers is given
in the following table:
.
1 2 . . 3 4 5 6 7 8 9 10
n
. .
p n 2 3 . 5 7 11 13 17 19 23 29
. .
H n 2 1 . 1 2 3 7 10 13 21 26
11 12 13 14 15 16 17 18 19 20
n
p n 31 37 41 43 47 53 59 61 67 71
H n 33 53 80 127 193 254 355 527 764 1149
Conjecture. The sequence {H n } is monotonic from H 3 onward.
This conjecture was contributed by one of the authors to an article en-
titled “Numbers Count” in the journal Personal Computer World and was
published in June 1991. Several readers checked its validity on computers,
but none of them found it to be false. The article is a regular one for com-
puter buffs and is conducted by Mike Mudge, a former colleague of the
author.
Exercise. Prove or refute the conjecture analytically.

5.8.6 Bordered Yamazaki–Hori Determinants — 2
A bordered determinant W of order (n+1) is defined in Section 4.10.3 and
is evaluated in Theorem 4.42 in the same section. Let that determinant be
denoted here by W n+1 and verify the formula

2
n 2
W n+1 = − K n (x − 1) n(n−1) {(x +1) − (x − 1) }
n
4
for several values of n. K n is the simple Hilbert determinant.
Replace the last column of W n+1 by the column

T
135 ··· (2n − 1) •