a30 := primorial(42681)*primorial(151)*primorial(29)*primorial(11) *primorial(7)*primorial(5)*primorial(4)*(primorial(3)^2) *(primorial(2)^4)*(primorial(1)^9); ? 10000*Qr(a30)%370 = 9999.10909330009471428528222805072598431749321026349669841 Qr(W) = sigma(W)/(harmonic(W)+log(harmonic(W))*exp(harmonic(W))) harmonic(Z)=Euler+psi(Z+1) primorial(W)=prod(Z=1,W,prime(Z))
Month: June 2013
About a “highly composite” number called “a8″
a8 = primorial(3302)*primorial(52)*primorial(14) *primorial(7)*(primorial(4)^2)*primorial(3) *(primorial(2)^4)*(primorial(1)^7); sigma(a8)/(H(a8)+log(H(a8))*exp(H(a8))) 0.999531374987652061429003800216023447038030443823248117806354
About a 12025-digit “highly composite” number called “a4”
I’ve been looking for whole numbers ‘n’ for whichthe sum of divisors function sigma(n) has a valuethat is “extremely large” relative to n. It can be shown that: limsup n→∞ σ(n)/[n log(log(n)) ] = e^gamma . (Grönwall, 1913) where sigma(n) is the sum of all divisors of the positive integer n. I got a 12,025-digit… Continue reading About a 12025-digit “highly composite” number called “a4”
About an 8307-digit “highly composite” number
The number ‘new’ below has 8307 digits .It is the product of 7 primorial numbers greater than or equal to 30, repetitionincluded in the count, and the two “high-powers” 2^10 and 3^3 ; So, new := (2^10)*(3^3)*P(3)*P(3)*P(4)*P(6)*P(12)*P(42)*P(2170) , where P(k) is the product of the ‘k’ first primes.For example, P(3) = 30 and P(4) =… Continue reading About an 8307-digit “highly composite” number
PARI/gp code related to RH criterion, Lagarias: sigma_1(n) asymptotics
The essential PARI/gp code to check my computation is below: ? primorial(1)%462 = 2? primorial(1)^6%463 = 64? m16%464 = m16? \uharmonic = (Z)->Euler+psi(Z+1) primorial = (W)->prod(X=1,W,prime(X)) ? Qr(W) = sigma(W,1)/(harmonic(W)+log(harmonic(W))*exp(harmonic(W)))%465 = (W)->sigma(W,1)/(harmonic(W)+log(harmonic(W))*exp(harmonic(W)))? m16%466 = m16? m16 = primorial(1614)*primorial(37)*primorial(11)*primorial(6)*primorial(4)*primorial(3)*primorial(2)^4*primorial(1)^6 ;? Qr(m16)%468 = 0.99923459761253613955435440267376833156?
Factoring highly composite number as primo. product, Result
? m16 = primorial(1614)*primorial(37)*primorial(11)* primorial(6)*primorial(4)*primorial(3)* primorial(2)^4*primorial(1)^6 ; ? big – m16%245 = 0? ==== done! Ok.
Factoring the Big number earlier as primorials product, Step 2
? (big/primorial(1614))/primorial(37)%233 = 3147213746118215854080000? factor((big/primorial(1614))/primorial(37))%234 = [2 14] [3 8] [5 4] [7 3] [11 2] [13 2] [17 1] [19 1] [23 1] [29 1] [31 1] ?
Factoring the Big number earlier as primorials product, Step 1
? big/primorial(1614)%230 = 111333211833855100982517905205279871065361695206259545187771188215857484030953369600000? factor(big/primorial(1614))%231 = [2 15] [3 9] [5 5] [7 4] [11 3] [13 3] [17 2] [19 2] [23 2] [29 2] [31 2] [37 1] [41 1] [43 1] [47 1] [53 1] [59 1] [61 1] [67 1] [71 1] [73 1] [79 1] [83 1] [89 1] [97… Continue reading Factoring the Big number earlier as primorials product, Step 1
a large abundant number, (think ~~= “supercolossal”)
From the PARI/gp screen: ? factor(b[11]*6*127*131*137*29*13*139*149*31*151*157*13577*13591*13597*13613*13619*13627*13633*13649*13669)%221 = [2 16] [3 10] [5 6] [7 5] [11 4] [13 4] [17 3] [19 3] [23 3] [29 3] [31 3] [37 2] [41 2] [43 2] [47 2] [53 2] [59 2] [61 2] [67 2] [71 2] [73 2] [79 2] [83 2] [89 2]… Continue reading a large abundant number, (think ~~= “supercolossal”)
Article explaining dynamically linked executables in Linux
Article explaining dynamically linked executables in Linux
I knew that it’s often the case that an executable (a program) needs “shared objects” or *so* files to work properly. The article introduces us to the ldd command, which tells us which library objects ( or *so* files) are required by a given Linux executable. This is very convenient. They also say something about dynamic library configuration, and ldconfig, which is very useful.
probable primes, degree 1024 polynomial
I’ve become intrigued by statistics of integer polynomial evaluations (large integers) with a fixed polynomial of a high degree, such as 1024, obtained by iterating 5 times over a degree 5 polynomial. In the case at hand, there are about 8 times as many probable primes in the polynomial values for the indeterminate X ranging… Continue reading probable primes, degree 1024 polynomial
Long story at the New Yorker
The Secret Sharer, by Jane Mayer, May 23 2011