some small improvements in English

chapters/Erste_Bsp.qmd

@@ -12,7 +12,7 @@ using InteractiveUtils
 # First Small Programs
 
 
-## What Should One Be Able to Do for Programming?
+## What Skills Are Needed for Programming?
 
 
 - **Thinking in algorithms:**  
@@ -30,7 +30,7 @@ using InteractiveUtils
 
 These cannot all be mastered at once. Be patient with yourself and 'play' with the language. 
 
-The following should serve as a small inspiration.
+The following examples should serve as inspiration.
 
 
 ## Project Euler
@@ -51,7 +51,7 @@ Aufg. 92
 :::
 
 ::: {.callout-note icon=false .titlenormal}
-## Project Euler Problem No 1
+## Project Euler Problem No. 1
 
 If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
 
@@ -65,7 +65,7 @@ Find the sum of all the multiples of 3 or 5 below 1000.
 
 2. Implementation:
 
-    How does Julia provide the remainder of integer division? Functions like this are typically called `rem()` (for *remainder*) or `mod()`. [Look it up in the documentation](https://docs.julialang.org/en/v1/base/math/#Base.rem) or try `?rem` and `?mod` in the Julia REPL shows that there are both. The difference lies in the treatment of negative integers. For natural numbers `n,m`, `mod(n,m)` and `rem(n,m)` give the same result, and the latter also has the infix form `n % m`.       
+    How does Julia provide the remainder of integer division? Functions like this are typically called `rem()` (for *remainder*) or `mod()`. You can look it up in the documentation or try `?rem` and `?mod` in the Julia REPL. Both functions exist; the difference lies in the treatment of negative integers. For natural numbers `n,m`, `mod(n,m)` and `rem(n,m)` give the same result, and the latter also has the infix form `n % m`.
      
     How does one test a sequence of values and sum them up? There is a standard pattern: `for` loop and accumulator variable:  
 
@@ -103,7 +103,7 @@ sum([i for i in 1:999 if i%3==0||i%5==0])
 
 
 ::: {.callout-note icon=false .titlenormal}
-## Project Euler Problem No 92
+## Project Euler Problem No. 92
 A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before.
 
 For example,
@@ -195,7 +195,7 @@ end
 println("$c numbers below ten million arrive at 89.")
 ```
 
-Numbers for which the iterative square digit sum calculation ends at 1 are called [happy numbers](https://en.wikipedia.org/wiki/Happy_number), and we have just calculated the number of sad numbers less than 10,000,000.
+Numbers for which the iterative square digit sum calculation ends at 1 are called [happy numbers](https://en.wikipedia.org/wiki/Happy_number); all other numbers eventually reach the 89-cycle.
 
 Here is the complete program again: