exercises were actually on chap0 so far

Author: shyba

math/how_to_prove_it/chap0.md

@@ -0,0 +1,93 @@
+1. a) 3 as factor, so 2³-1, which give us 7. the other is 32767/7=4681
+b) 2³²⁷⁶⁷-1, so 7 is a factor. 2⁷-1=127
+
+2) Suppose n is an integer greater than 1, 3^n-1 is even.
+
+3)
+a) 2,3,5,7 so
+M=2x3x5x7+1=211
+
+b) 2,5,11
+M=2x3x5x7x11+1=2311
+
+4)
+```python
+isprime = lambda x: all(x%y!=0 for y in range(2,x))
+
+primes = []
+for n in range(10000):
+    if len(primes) == 5: break
+    if isprime(n): primes.clear()
+    else: primes.append(n)
+
+primes
+[24, 25, 26, 27, 28]
+```
+
+5)
+from the table, 3 and 7 are valid options, so 28 and 8128 as perfect
+
+as an extra exercise, continuing the table:
+
+n | isPrime | 2^n-1 | is 2^n-1 prime? |
+--|---------|-------|-----------------|
+11|   yes   | 2047  |       no        |
+12|   no    | 4095  |       no        |
+13|   yes   | 8191  |       yes       |
+14|   no    | 16383 |       no        |
+15|   no    | 32767 |       no        |
+16|   no    | 65535 |       no        |
+17|   yes   | 131071|       yes       |
+
+candidates are 8191 and 131071
+from Euclid, 2^(n-1)*(2^n-1) is perfect, so
+33550336 and 8589869056
+
+```python
+perfect = lambda x: 2**(x-1)*(2**x-1)
+isperfect = lambda x: x == sum(i for i in range(1, x) if x%i==0)
+for x in range(1,13):
+     print(x, isprime(x), 2**x-1, isprime(2**x-1), perfect(x), isperfect(perfect(x)))```
+```
+
+output:
+```
+1 True 1 True 1 False
+2 True 3 True 6 True
+3 True 7 True 28 True
+4 False 15 False 120 False
+5 True 31 True 496 True
+6 False 63 False 2016 False
+7 True 127 True 8128 True
+8 False 255 False 32640 False
+9 False 511 False 130816 False
+10 False 1023 False 523776 False
+11 True 2047 False 2096128 False
+12 False 4095 False 8386560 False
+```
+
+6)
+```python
+for n in range(1, 30000):
+    if isprime(n) and isprime(n+2) and isprime(n+4):
+        print(n, n+2, n+4)
+```
+```
+1 3 5
+3 5 7
+```
+up to 30,000 those are the only 2 triplets
+
+7)
+```python
+small_div_sum = lambda x: sum(n for n in range(1, x) if x%n==0)
+amicables = lambda x, y: small_div_sum(x) == y and small_div_sum(y) == x
+amicables(220, 284)
+amicables(20, 30)
+```
+
+--->
+```
+True
+False
+```

math/how_to_prove_it/chap1.md

@@ -1,93 +0,0 @@
-1. a) 3 as factor, so 2³-1, which give us 7. the other is 32767/7=4681
-b) 2³²⁷⁶⁷-1, so 7 is a factor. 2⁷-1=127
-
-2) Suppose n is an integer greater than 1, 3^n-1 is even.
-
-3)
-a) 2,3,5,7 so
-M=2x3x5x7+1=211
-
-b) 2,5,11
-M=2x3x5x7x11+1=2311
-
-4)
-```python
-isprime = lambda x: all(x%y!=0 for y in range(2,x))
-
-primes = []
-for n in range(10000):
-    if len(primes) == 5: break
-    if isprime(n): primes.clear()
-    else: primes.append(n)
-
-primes
-[24, 25, 26, 27, 28]
-```
-
-5)
-from the table, 3 and 7 are valid options, so 28 and 8128 as perfect
-
-as an extra exercise, continuing the table:
-
-n | isPrime | 2^n-1 | is 2^n-1 prime? |
---|---------|-------|-----------------|
-11|   yes   | 2047  |       no        |
-12|   no    | 4095  |       no        |
-13|   yes   | 8191  |       yes       |
-14|   no    | 16383 |       no        |
-15|   no    | 32767 |       no        |
-16|   no    | 65535 |       no        |
-17|   yes   | 131071|       yes       |
-
-candidates are 8191 and 131071
-from Euclid, 2^(n-1)*(2^n-1) is perfect, so
-33550336 and 8589869056
-
-```python
-perfect = lambda x: 2**(x-1)*(2**x-1)
-isperfect = lambda x: x == sum(i for i in range(1, x) if x%i==0)
-for x in range(1,13):
-     print(x, isprime(x), 2**x-1, isprime(2**x-1), perfect(x), isperfect(perfect(x)))```
-```
-
-output:
-```
-1 True 1 True 1 False
-2 True 3 True 6 True
-3 True 7 True 28 True
-4 False 15 False 120 False
-5 True 31 True 496 True
-6 False 63 False 2016 False
-7 True 127 True 8128 True
-8 False 255 False 32640 False
-9 False 511 False 130816 False
-10 False 1023 False 523776 False
-11 True 2047 False 2096128 False
-12 False 4095 False 8386560 False
-```
-
-6)
-```python
-for n in range(1, 30000):
-    if isprime(n) and isprime(n+2) and isprime(n+4):
-        print(n, n+2, n+4)
-```
-```
-1 3 5
-3 5 7
-```
-up to 30,000 those are the only 2 triplets
-
-7)
-```python
-small_div_sum = lambda x: sum(n for n in range(1, x) if x%n==0)
-amicables = lambda x, y: small_div_sum(x) == y and small_div_sum(y) == x
-amicables(220, 284)
-amicables(20, 30)
-```
-
---->
-```
-True
-False
-```