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Pythagorean Plus One (Posted on 2004-05-17) Difficulty: 3 of 5
A "Pythagorean Plus One" triple can be defined as any three distinct integers a, b, c, such all three of these are one more than a perfect square, and also a times b equals c. What is the lowest value of c possible?

  Submitted by Gamer    
Rating: 2.5000 (2 votes)
Solution: (Hide)
All the numbers need to be distinct, so 0 can't equal a, b, or c. The lowest other possibility is a=1, b=a+1=2, which results in a value of c=3. A way to prove that b can be a+1 is posted here:

(a^2+1)((a+1)^2+1)=
(a^2)(a^2+2a+1)+(a^2)+(a^2+2a+1)+1=
(a^4+2a^3+a^2+a^2+a^2+a^2+2a+1)+1=
(a^4+ax^3+ax^2+ax+1)+1=
(a^2+a+1)^2+1.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
The official solution is wrong:Ady TZIDON2010-02-23 04:16:13
Euan seems to be wrong...Penny2004-06-05 20:21:35
re: People seem to be wrong...Tristan2004-06-04 18:11:26
People seem to be wrong...Euan2004-06-04 06:37:27
re: solutionRichard2004-05-22 21:24:38
Some ThoughtssolutionGururaj2004-05-22 04:43:22
no subjectGururaj2004-05-21 08:32:02
re: No Subjectperplexus2004-05-20 22:06:25
No Subjectperplexus2004-05-20 22:02:31
Pythagorean Minus OnePenny2004-05-17 18:41:34
Yet another variation...Penny2004-05-17 18:30:09
re: Maybe a really smart floobler.....Richard2004-05-17 14:54:34
Maybe a really smart floobler.....Penny2004-05-17 13:27:00
re: Hen's eggsDJ2004-05-17 12:39:36
Hen's eggsJer2004-05-17 12:26:56
All odd Pythagorean Plus One triplesPenny2004-05-17 10:04:12
The smallest 4-digit solution.....Penny2004-05-17 09:40:33
Solutionsolutionderek2004-05-17 09:23:27
SolutionSolutionPenny2004-05-17 09:11:09
Some ThoughtsSolution??Oskar2004-05-17 08:50:36
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