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# kg583 / TINT

Complete repository for the TI Number Theory package

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# TINT Package v1.2 Info

TINT, or TI Number Theory, is a package of lists and programs designed for number theoretic computation and analysis on the TI-83+ series of calculators. These programs are designed to be used as subprograms for larger projects, and are optimized for numbers less than 10^14. TINT's main feature is the precomputation of many arithmetic properties of a given input number stored for later use (e.g. number of divisors, totient function, etc.).

To install, simply download and open the group TINT, or you may download/copy individuals programs directly. All TINT programs are denoted by θNT followed by 2 or 3 characters. The following variables and lists are defined for an input N after executing the program θNTN. Most programs also require the factorization of N to be defined via θNTN (exceptions are noted below).

# Updated in v1.1

• Removed all remainder( commands to allow support for older OS's
• Added primality/probable prime tests (see documentation for θNTPT)
• Added Lucas sequence generation (see documentation for θNTLS)
• Added more numerical properties to θNTIS
• Adjusted variable allocation for less shuffling (most inputs that are "bases" are now θ instead of Q)
• Better edge case support
• General bug fixes

# Updated in v1.2

• Added Zeda's fast modular exponentiation algorithm (see documentation for θNTMZ)

# Vars

## Properties

• W: # of distinct prime divisors
• O: # of prime divisors
• C: Compositeness
• T: # of Divisors
• S: Sum of Divisors
• A: Aliquot sum
• B: Abundance
• H: Totient
• L: Liouville function
• U: Mobius function
• D: Arithmetic derivative
• G: Log arithmetic derivative

## Inputs

• N: Number
• M: Arbitrary whole number (usually a modulus)
• P: Arbitrary whole number (usually a prime)
• Q: Arbitrary whole number
• K: Input number (usually small)
• θ: Input number (usually a base)

• X: Bound
• Y: Loop var
• Z: Counter

• E: Empty var
• F: Empty var
• I: Empty var
• J: Empty var
• V: Empty var

## Lists

• P: Prime factors of N
• A: Prime multiplicities of N
• B: A+1 (useful for divisor-related queries)
• D: Divisors of N
• E: Divisor multiplicities of N
• U: Unitary divisors of N (those divisors with multiplicity 1)

• P100: All primes < 100
• P1000: All primes < 1000

# Programs

Return is Ans unless otherwise stated; if a return is also stored in a variable, it is denoted in parentheses

• NTCL: List of integers between 1 and N coprime to N --> L1; v1.1
• NTCM: Carmichael's function of N
• NTCQ: Ramanujan's sum of N base Q
• NTCR: Core of N mod M
• NTDK: Sum of Kth powers of divisors of N (K=0 <--> # of Divisors, K=1 <--> Sum of Divisors)
• NTET: Euler's totient function of N (w/o known factorization); v1.1
• NTFP: Find the smallest prime factor of N (w/o known factorization)
• NTGCD: GCD of L1
• NTIS: If N satisfies property; input the property as a two-letter string from the list below in Ans; tests may not always return 1 for a positive result
• "CM": Carmichael
• "LC": Lucas-Carmichael
• "KP": K-Perfect (K=2 <--> Perfect)
• "GI": Giuga
• "SF": Squarefree
• "KS": K-Smooth
• "KR": K-Rough
• "PW": Perfect power
• "AB": Abundant
• "PF": Powerful
• "UN": Unusual
• "RF": Refactorable
• "SP": Semiprime
• "NH": Nonhypotenuse
• "AP": Almost perfect
• "KH": K-Hyperperfect
• "HP": Hemiperfect
• "BL": Blum
• "RG": Regular
• "HD": Harmonic divisor
• "AR": Arithmetic
• "PP": Primary pseudoperfect
• "KA": K-Almost prime (K=1 <--> Prime, K=2 <--> Semiprime, K=3 <--> Sphenic); v1.1
• "CK": Coprime to K; v1.1
• "PR": N has primitive roots; v1.1
• "θR": θ is a primitive root modulo N; v1.1
• "SH": Sphenic; v1.1
• NTJK: Jordan's totient of N base K
• NTLCM: LCM of L1
• NTLI: Intersection of L1 and L2 --> L3
• NTLJ: Kronecker symbol (generalized Legendre/Jacobi symbol) of N base θ (--> Z)
• NTLS: Establish Lucas sequences U(P,Q) & V(P,Q) --> u & v (respectively); v1.1
• NTMO: Multiplicative order of θ mod M
• NTMU: Mobius function of N (w/o known factorization)
• NTMX: θ^K mod M (modular exponentiation algorithm, preserves θ and K)
• NTMZ: θ^K mod M (fast modular exponentiation algorithm, destroys θ and K); v1.2
• NTN: TINT data initialization for N
• NTPF: Prime factorization of N; does not preserve N
• NTPG: List of primes up to X --> L1
• NTPN: Generate next prime given all previous primes in L1
• NTPT: If N is prime (w/o known factorization); v1.1: input the test as a two-letter string from the list below in Ans; a trial divison by 2 is always performed; tests may not always return 1 for a positive result
• "TD": Trial division (all odd integers from 3 to √N)
• "AD": Accelerated trial division (3, all integers ±1 mod 6 from 5 to √N)
• "PD": Prime trial division (all primes from 2 to √N)
• "Fθ": Fermat probable prime test base θ
• "FR": Fermat probable prime with random base
• "Sθ" or "MR": Strong probable prime test base θ (also called the Miller-Rabin primality test)
• "SR": Strong probable prime test with random base
• "Eθ": Euler probable prime test base θ
• "ER": Euler probable prime test with random base
• "LU": Lucas probable prime test for parameters P & Q
• "LS": Strong Lucas probable prime test for parameters P & Q
• "LE": Extra strong Lucas probable prime test for parameters P & Q
• "BP": Baillie-PSW primality test
• "FB": Frobenius probable prime test for parameters P & Q
• "FI": Fibonacci probable prime test
• "PL": Pell probable prime test
• NTRL: List of primitive roots modulo N --> L1; v1.1
• NTVP: Multiplicity of P in N (--> Z)

Have any questions? Found a bug? Contact kg583 on TI-Basic Developer or Cemetech.

Complete repository for the TI Number Theory package

## Releases 1

TI Number Theory (TINT) Latest
Feb 1, 2019

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