WXML Spring 2016

Number Theory and Noise

Quadratic residues

All files in this section created by Christine Wolf

Let m be a positive integer. Define a set of positive integers A by saying that n is in A iff n is a quadratic residue modulo m.

The following are the sound files resulting different moduli m.

Powers of 2

• Quadratic residues, modulo 8 (10 sec)
• Quadratic residues, modulo 16 (10 sec)
• Quadratic residues, modulo 32 (10 sec)
• Quadratic residues, modulo 64 (10 sec)
• Quadratic residues, modulo 128 (10 sec)
• Quadratic residues, modulo 256 (10 sec)
• Quadratic residues, modulo 512 (10 sec)
• Quadratic residues, modulo 1024 (10 sec)
• Quadratic residues, modulo 2048 (10 sec)
• Quadratic residues, modulo 4096 (10 sec)
• Quadratic residues, modulo 8192 (10 sec)
• Quadratic residues, modulo 16384 (10 sec)
• Quadratic residues, modulo 32786 (10 sec)
• Quadratic residues, modulo 65536 (10 sec)
• Quadratic residues, modulo 131072 (10 sec)

Powers of 3

• Quadratic residues, modulo 3 (10 sec)
• Quadratic residues, modulo 9 (10 sec)
• Quadratic residues, modulo 27 (10 sec)
• Quadratic residues, modulo 81 (10 sec)
• Quadratic residues, modulo 243 (10 sec)
• Quadratic residues, modulo 729 (10 sec)
• Quadratic residues, modulo 2187 (10 sec)
• Quadratic residues, modulo 6561 (10 sec)
• Quadratic residues, modulo 19683 (10 sec)
• Quadratic residues, modulo 59049 (10 sec)

Powers of 5

• Quadratic residues, modulo 5 (10 sec)
• Quadratic residues, modulo 25 (10 sec)
• Quadratic residues, modulo 125 (10 sec)
• Quadratic residues, modulo 625 (10 sec)
• Quadratic residues, modulo 3125 (10 sec)
• Quadratic residues, modulo 15625 (10 sec)
• Quadratic residues, modulo 78125 (10 sec)

Powers of 7

• Quadratic residues, modulo 7 (10 sec) spectrogram
• Quadratic residues, modulo 49 (10 sec) spectrogram
• Quadratic residues, modulo 343 (10 sec) spectrogram
• Quadratic residues, modulo 2401 (10 sec) spectrogram
• Quadratic residues, modulo 16807 (10 sec) spectrogram

Powers of 11

• Quadratic residues, modulo 11 (10 sec) spectrogram
• Quadratic residues, modulo 121 (10 sec) spectrogram
• Quadratic residues, modulo 1331 (10 sec) spectrogram
• Quadratic residues, modulo 14641 (10 sec) spectrogram

Moduli of the form (11)(7ⁿ)

• Quadratic residues, modulo 77 (10 sec) spectrogram
• Quadratic residues, modulo 539 (10 sec) spectrogram
• Quadratic residues, modulo 3773 (10 sec) spectrogram
• Quadratic residues, modulo 26411 (10 sec) spectrogram

modulo 40000 to 40010

• Quadratic residues modulo 40000 (30 sec.)
• Quadratic residues modulo 40001 (30 sec.)
• Quadratic residues modulo 40002 (30 sec.)
• Quadratic residues modulo 40003 (30 sec.)
• Quadratic residues modulo 40004 (30 sec.)
• Quadratic residues modulo 40005 (30 sec.)
• Quadratic residues modulo 40006 (30 sec.)
• Quadratic residues modulo 40007 (30 sec.)
• Quadratic residues modulo 40008 (30 sec.)
• Quadratic residues modulo 40009 (30 sec.)
• Quadratic residues modulo 40010 (30 sec.)

Longer sounds and simulated sequences

All files in this section created by Xinwei Fan

• Five minutes of prime noise (primes less than 13.3 million) (300 sec.)

• Five minutes of abundant numbers (abundant numbers less than 13.3 million) (300 sec.)
• Simulated abundant numbers: multiples of 6, plus randomly included integers, maintaining asymptotic density
• Simulated abundant numbers: multiples of 6 and 20, plus randomly included integers, maintaining asymptotic density

Digit-related sequences

All files in this section created by Elana Lessing

Integers with certain digit sums

• Integers with digit sum divisible by 2 (23 sec)
• Integers with digit sum divisible by 3 (23 sec)
• Integers with digit sum divisible by 4 (23 sec)
• Integers with digit sum divisible by 5 (23 sec)
• Integers with digit sum divisible by 6 (23 sec)
• Integers with digit sum divisible by 7 (23 sec)
• Integers with digit sum divisible by 8 (23 sec)
• Integers with digit sum divisible by 9 (23 sec)
• Integers with digit sum divisible by 10 (23 sec)

Integers with certain digit products

• Integers with digit product (ignoring zeroes) less than 5 (23 sec)
• Integers with digit product (ignoring zeroes) less than 40 (23 sec)
• Integers with digit product (ignoring zeroes) less than 50 (23 sec)
• Integers with digit product (ignoring zeroes) less than 100 (23 sec)

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