Radcliffe

import csv
import requests
from collections import Counter
import json

url = 'https://raw.githubusercontent.com/soehms/database_knotinfo/main/database_knotinfo/csv_data/knotinfo_data_complete.csv'

with requests.Session() as session:
    download = session.get(url)
    decoded = download.content.decode('utf-8')

Radcliffe

"""
In 1769, Euler conjectured that it was impossible to express an nth power as the sum of fewer than
n nth powers of positive integers. For example, the sum of two cubes is not a cube, the sum of three
fourth powers is not a fourth power, and so on.

In 1966, L. J. Lander and T. R. Parkin found a counterexample: 27^5 + 84^5 + 110^5 + 133^5 = 144^5.
They performed a computer search using a CDC 6600. This was a massive computation for its time, but
it's easy for modern computers.

The following Python script searches for positive integer solutions to a^5 + b^5 + c^5 + d^5 = e^5

Radcliffe

from fractions import Fraction
import sys

"""
Represent a positive rational number as the sum of exactly k unit fractions, distinct or not.
This function returns the first solution in lexicographic order, represented as a list of denominators.
If no solution exists, the function returns None.

We can use the following observation to limit the search space to a finite number of possibilities.
Suppose that r = 1/n_1 + 1/n_2 + ... + 1/n_k, where n_1 ≤ n_2 ≤ ... ≤ n_k.

Radcliffe

Radcliffe

id,content_type,category_id,category_name,grade_id,grade_name,item,description
CCSS.ELA-LITERACY.L.K.1,ELA-LITERACY,L,Language,K,Kindergarten,1,Demonstrate command of the conventions of standard English grammar and usage when writing or speaking.
CCSS.ELA-LITERACY.L.K.1.a,ELA-LITERACY,L,Language,K,Kindergarten,1a,Print many upper- and lowercase letters.
CCSS.ELA-LITERACY.L.K.1.b,ELA-LITERACY,L,Language,K,Kindergarten,1b,Use frequently occurring nouns and verbs.
CCSS.ELA-LITERACY.L.K.1.c,ELA-LITERACY,L,Language,K,Kindergarten,1c,"Form regular plural nouns orally by adding /s/ or /es/ (e.g., dog, dogs; wish, wishes)."
CCSS.ELA-LITERACY.L.K.1.d,ELA-LITERACY,L,Language,K,Kindergarten,1d,"Understand and use question words (interrogatives) (e.g., who, what, where, when, why, how)."
CCSS.ELA-LITERACY.L.K.1.e,ELA-LITERACY,L,Language,K,Kindergarten,1e,"Use the most frequently occurring prepositions (e.g., to, from, in, out, on, off, for, of, by, with)."
CCSS.ELA-LITERACY.L.K.1.f,ELA-LITERACY,L,Language,K,Kindergarten,

Radcliffe

"""
The number 10 has a special property. When 10 is added to the sum 
of its divisors, the result is 28, which is a perfect number.

    10 + sigma(10) = 10 + 1 + 2 + 5 + 10 = 28

Are there other perfect numbers that can be expressed as the sum 
of a number and the sum of its divisors?