80's Music and Math

Just read a link about the song below on Dave Richeson's "Division by Zero" blog. First, listen and try to figure out what is the cool math in the song...


Ok, Dave's web site is in response to a previous post about the Twin Primes Conjecture, and here is what he found when he checked it out on Wolfram Alpha

"Type 8675309 into WolframAlpha and it says: “Jenny’s phone number.” (Har, har, har, we all know that song.)

At the top of the page it says: Assuming “8675309″ is a phrase | Use as a number instead.

Click on that link and you’ll see the following interesting facts about 8675309.

1. 8675309 is a prime.
2. 8675309 is a twin prime (8675311 is also prime).
3. 8675309 is the hypotenuse of a (primitive) Pythagorean triple: 8675309² = 2460260²+8319141².

That’s so cool! Who knew?!?!"

Well, Not me, Dave, but I'm willing to help spread the word..
by the way, What is the smallest number that would meet all three conditions above, that is it is a twin prime and it is also the hypotenuse of a primitive Pythagorean triple.... Ok, way to easy ... what is the NEXT smallest?

The obvious question, is it really a phone number???

Yep, and here is the straight scoop from the folks at Snopes
To add a little statistical flavor..."In a project he called Jenny, are you there?, Dan Stecz, a New Jersey man, called all the 867-5309s in every area code within North America and found that nearly all the numbers were not in service. A handful of the numbers did refer to Jenny, however, and some even played bits of the song on their answering machine greeting."[wikipedia]

In 2015 I came across this while searching Google, ""Dexter" writes that if you use Vonage as your phone company, when making a call to any "867-5309" number, they play the chorus of "Jenny" before completing ..."  Have not verified that, but
Hey, Dave, Now THAT's Cool