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 {pn} and define a

Hessenberg determinantHn(Section 4.6) as follows:

Hn=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

p 1 p 2 p 3 p 4 ···

1 p 1 p 2 p 3 ···

1 p 1 p 2 ···

1 p 1 ···

··· ···

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

This determinant satisfies the recurrence relation

Hn=

n− 1

∑

r=0

(−1)

r

pr+1Hn− 1 −r,H 0 =1.

A short list of primes and their associated Hessenberg numbers is given

in the following table:

n 12

.345678910

pn 23

.57111317192329

Hn 21

. 1 2 3 7 10 13 21 26

n 11 12 13 14 15 16 17 18 19 20

pn 31 37 41 43 47 53 59 61 67 71

Hn 33 53 80 127 193 254 355 527 764 1149

Conjecture.The sequence{Hn}is monotonic fromH 3 onward.

This conjecture was contributed by one of the authors to an article en-

titled “Numbers Count” in the journalPersonal Computer Worldand 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 determinantWof 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 byWn+1and verify the formula

Wn+1=−

Kn

4

(x

2

−1)

n(n−1)

{(x+1)

n

−(x−1)

n

}

2

for several values ofn.Knis the simple Hilbert determinant.

Replace the last column ofWn+1by the column

[

135 ···(2n−1)•

]T