Testwiki:Reference desk/Archives/Mathematics/2015 August 5

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August 5

Why are large integers made of a string of 7s so easy to factor???

The new HP Prime calculator can find the prime factors of a integer...if one puts in a long strong of 7s, it can factor it easily?? Why...When I use the software version of the HP Prime calculator, I can put in an integer made up of 50 7s in a row, and it is factored easily. If I put in a random 50 digit number, not so easily, and it may not be able to factor it at all...any math experts know why? Curious... — Preceding unsigned comment added by 208.118.28.7 (talk) 13:48, 5 August 2015 (UTC)

Such a number is seven times a repunit. These can be at least partially factored using cyclotomic polynomials, although I do not know what algorithm the calculator uses. Sławomir
Biały 14:00, 5 August 2015 (UTC)

I asked a similar question a few weeks ago, but about numbers like $10^{67}+1$. You may view a sequence-of-sevens number as being of the form $(10^n-1)*7/9$, and depending on the value of 'n' the factorization will be helped by the polynomial factorization of $x^n-y^n$, with x=10 and y=1. HTH, Robinh (talk) 20:10, 5 August 2015 (UTC)

minimum value of y=x^x-x*x question

Wolfram Alpha says that the minimum value of y=x^x-x*x is at about x=1.62028 and has a value of approximately -.439587. Does either the x or the y have anything approaching a nice form?— Preceding unsigned comment added by Naraht (talk • contribs)

Of course not. --JBL (talk) 20:38, 5 August 2015 (UTC)

"Of course" is a strong word...

But yeah, the inverse symbolic calculator (aka "the new Plouffe's Inverter") got nothing. -- Meni Rosenfeld (talk) 10:49, 6 August 2015 (UTC)

The equation in question is a trancendental equation, likely to have one of three behaviors: there might be (1) an obvious, easily checkable solution; (2) a "standard" function whose definition is "the value of a solution to the equation in question" (a la Lambert's W function); or (3) nothing to say. In this case, it's easy to rule out (1) and (2). --JBL (talk) 14:39, 6 August 2015 (UTC)

I don't know if the solution has anything approaching a nice form, but the nice form $x=\phi$ seems to approach the real solution. (Where $\phi$ is the Golden Ratio). 129.234.186.11 (talk) 12:06, 6 August 2015 (UTC)

There is no more connection between these two numbers than there is between any two random real numbers that happen to agree to one decimal place. --JBL (talk) 14:39, 6 August 2015 (UTC)