[math-fun] a Cauchy equation question
g(x+y)=g(x)g(y) x,y are belong to R (BUT there is no continuous condition g may not be a continuous function) how to prove that if g(1)>1 then g is increasing if g(1)=1 then g is a constant function if g(1)<1 then g is deceasing ----------------------------------------------------------------- §Ûªñ¸ô¡I¡IEmail³B³B¦¬¡I §K¶O¤U¸üYahoo!©_¼¯±¶®|¦C http://tw.companion.yahoo.com/
--- ±i®a»ô <chiachichang1123@yahoo.com.tw> wrote:
g(x+y)=g(x)g(y) x,y are belong to R
(BUT there is no continuous condition
g may not be a continuous function)
how to prove that
if g(1)>1 then g is increasing
if g(1)=1 then g is a constant function
if g(1)<1 then g is deceasing
From the functional equation, either g(x) vanishes identically or g(0)=1. Given g(1), the functional equation determines g(n) for integer n. If we make the additional assumption that g is everywhere nonnegative, then g(r) is determined on the rationals r. And, on the rationals, g satisfies the above monotonicity conditions. But this is as far as one can go without some way to relate the irrationals to the rationals. Continuity is one way to do this, and then one gets
g(x) = exp(x log(g(1))). The reals R are a vector space over the rationals Q. Using the axiom of choice, let B be a basis of R over Q. Then for each b in B, G(b) may be arbitrarily chosen. Gene __________________________________ Do you Yahoo!? Yahoo! Tax Center - File online by April 15th http://taxes.yahoo.com/filing.html
Let the nth Euclid number E_n be defined as 1 + (p_1 * ... * p_n) (aka 1 + the nth "primorial"), where p_n is the nth prime number. Neil Sloane's EIS sequence A006862 lists the first few n for which E_n is prime: 1,2,3,4,5,11,75,171,172,384,457,616,643,.... Some questions: 1. Are there infinitely many prime (resp. composite) E_n ??? 2. Is there a nice asymptotic expression for the number of E_n < x ??? 3. Same for prime (resp. composite) E_n ??? --Dan Daniel Asimov Visiting Scholar Mathematics Department University of California Berkeley, California
Little hope of answering Dan's questions in the foreseeable future. Could someone check this against UPINT3 A2 the beginning of which is quoted below. The primes listed at p# + 1 comprise sequence A005234. A reference to UPINT A2 would be in order. Their ranks comprise sequence A014545. I checked this against Abramowitz & Stegun, except for the last three entries. This is the sequence partially quoted by Dan. The sequence number he mentions, A006862, is not the same. ^^^^^^ Does anyone know of additional entries to any of these sequences? (I'm working on UPINT4 :-) R. ----------------- \usection{A2}{Primes connected with factorials.} \hGidx{factorial $n$} Are there infinitely many primes of the form $n!\pm1$ or of the form $p\#\pm1$, where $p\#$ is the product, \Gidx{primorial $p$}, of the primes $2\cdot3\cdot5\cdots p$ up to $p$. Discoveries since the second edition by Harvey Dubner and others have brought the lists to: $n!+1$ is prime for $n=1$, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380. $n!-1$ is prime for $n=3$, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480. $p\#+1$ is prime for $p=2$, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113. $p\#-1$ is prime for $p=3$, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877. -------------------- On Tue, 13 Apr 2004, Dan Asimov wrote:
Let the nth Euclid number E_n be defined as 1 + (p_1 * ... * p_n) (aka 1 + the nth "primorial"), where p_n is the nth prime number.
Neil Sloane's EIS sequence A006862 lists the first few n for which E_n is prime:
1,2,3,4,5,11,75,171,172,384,457,616,643,....
Some questions:
1. Are there infinitely many prime (resp. composite) E_n ???
2. Is there a nice asymptotic expression for the number of E_n < x ???
3. Same for prime (resp. composite) E_n ???
--Dan
Daniel Asimov Visiting Scholar Mathematics Department University of California Berkeley, California
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