2. From numerical experimentation, it would appear (miraculously) that the value of (²) for odd primes is determined by the value of q( mod p). What about 2 ? The value of (3) certainly isn't determined by the value of p(mod2) - but maybe it is determined by the value of p(mod4) ? Is Podasip 8 helpful? Make a conjecture. Annexes: Podasip 8. Keep the notation from Podasip 7. If p is an odd prime and a = 0(modp), then... (a) ... a¹ = (-1) (modp) where N is the number of ek that are negative. (b) = (-1) M (modp) where M = 1 + €₂ + + €(p-1)/2- ...a Podasip 7. Suppose p is an odd prime and a # 0(modp). For each k = 1,2,..., ¹ define ek and rk by ak= €kTk (modp) where 0<< and Then those remainders 71, 72,..., 7(p-1)/2 are distinct. Р €k = ±1.
2. From numerical experimentation, it would appear (miraculously) that the value of (²) for odd primes is determined by the value of q( mod p). What about 2 ? The value of (3) certainly isn't determined by the value of p(mod2) - but maybe it is determined by the value of p(mod4) ? Is Podasip 8 helpful? Make a conjecture. Annexes: Podasip 8. Keep the notation from Podasip 7. If p is an odd prime and a = 0(modp), then... (a) ... a¹ = (-1) (modp) where N is the number of ek that are negative. (b) = (-1) M (modp) where M = 1 + €₂ + + €(p-1)/2- ...a Podasip 7. Suppose p is an odd prime and a # 0(modp). For each k = 1,2,..., ¹ define ek and rk by ak= €kTk (modp) where 0<< and Then those remainders 71, 72,..., 7(p-1)/2 are distinct. Р €k = ±1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:2. From numerical experimentation, it would appear (miraculously) that the value of() for odd
primes is determined by the value of q( mod p). What about 2 ? The value of (3) certainly isn't determined
by the value of p(mod2) - but maybe it is determined by the value of p(mod4) ? Is Podasip 8 helpful?
Make a conjecture.
Annexes:
Podasip 8. Keep the notation from Podasip 7. If p is an odd prime and a = 0(modp), then... (a)
a²¹ = (−1)ª (modp) where N is the number of € that are negative. (b) = (-1)M (modp)
2
by
where M = ₁ + €₂ + + €(p-1)/2-
Podasip 7. Suppose p is an odd prime and a ‡ 0(modp). For each k = 1,2, ...,
Р
ak Ekk (modp) where 0 <rk <
2
Then those remainders 7₁, 72,..., 7(p-1)/2 are distinct.
... a
and
€k = ±1.
P¹ define € and rk
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