God Preserves Kepler’s Great Treasure of Gold!
"It is impossible for a cube to be the sum of two [positive integer] cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."
Pierre de Fermat  conjectured in 1637 and proved in 1994 by Andrew Wiles
Proof of Fermat's “last theorem” purifies Kepler’s “measure of gold” so that the special place in God’s Mind for the 345 Pythagorean “parent of all triples” might shine forth delightfully in conjunction with Kepler’s other “great treasure” — the Golden Ratio!
Interrelationship of the 345 Pythagorean Triple and the Golden Ratio:
Consider the following proof from H. E. Huntley's The Divine Proportion, Dover, 1970
PQ / BP = φ
Let ABC be such a triangle with BC = 3, AC = 4 and AB = 5.
Let O be the foot of the angle bisector at B.
Draw a circle with center O and radius CO.
Extend BO to meet the circle at Q
Let P be the other point of intersection of BO with the circle.
Then PQ / BP = φ.
AO = 5/2 and CO = 3/2.
Thus the circle's radius r is 3/2.
By the Power of a Point Theorem,
BP·BQ = BC^{2}.
In other words, (BO  3/2)·(BO + 3/2) = 3^{2}.
From which, BO = 3√5/2. We thus find BP = 3(√5  1)/2.
And finally,
PQ / BP = 2·r / [3(√5  1)/2]
= 2 / (√5  1)
= 2 · (√5 + 1) / 4
= (√5 + 1) / 2 = φ
Thus: PQ / BP = (√5 + 1) / 2 = φ
The Tree of Primitive Pythagorean Triples
••• ••• •••
91 
60 
109 
187 
84 
205 
117 
44 
125 
299 
180 
349 
459 
220 
509 
165 
52 
173 
209 
120 
241 
273 
136 
305 
63 
16 
65 

105 
88 
137 
297 
304 
425 
207 
224 
305 
377 
336 
505 
697 
696 
985 
319 
360 
481 
275 
252 
373 
403 
396 
565 
133 
156 
205 

9 
40 
41 
105 
208 
233 
95 
168 
193 
57 
176 
185 
217 
456 
505 
175 
288 
337 
51 
140 
149 
115 
252 
277 
85 
132 
157 

Generation 3 
Generation 3 
Generation 3 

Matrix 

Matrix 

Matrix  
7 
24 
25 
55 
48 
73 
45 
28 
53 
39 
80 
89 
119 
120 
169 
77 
36 
85 
33 
56 
65 
65 
72 
97 
65 
72 
97 

Generation 2 
Generation 2 
Generation 2 

Matrix 

Matrix 

Matrix 

5 
12 
13 
21 
20 
29 
15 
8 
17 

Generation 1 
Generation 1 
Generation 1 

1 
2 
2 
1 
2 
2 
1 
2 
2 

2 
1 
2 
2 
1 
2 
2 
1 
2 

2 
2 
3 
2 
2 
3 
2 
2 
3 

Matrices 

3 
4 
5 

Original Parents 
In mathematics, a Pythagorean triple is a set of three positive integers a, b, and c having the property that they can be respectively the two legs and the hypotenuse of a right triangle, thus satisfying the equation a^{2} + b^{2} = c^{2}; the triple is said to be primitive if and only if a, b, and c share no common divisor. The set of all primitive Pythagorean triples has the structure of a rooted tree, specifically a ternary tree, in a natural way.
Each child is itself the parent of 3 more children, and so on. If one begins with primitive triple [3, 4, 5], all primitive triples will eventually be produced by application of [above] matrices. The result can be graphically represented as an infinite ternary tree.
Wikipedia
"Wisdom is proved right by her children" Luke 7:35
The numbers 5, 12, and 60 are the 'mitochondrial DNA organelles'
which trace the ancestry of all Pythagorean triples back to a first parent —
The 345 Triangle!
One side of every Pythagorean triple is divisible by 5.
The product of the two nonhypotenuse legs is always divisible by 12.
The largest number that always divides the product of all three sides is 60.
The age our universe can be determined utilizing the numbers:
5 12 60,
the Golden Ratio Ø,
1 Plank time = 5.391x1044 secs,
and the literal truth of Isaiah 40:12.
5 (Ø x 10 ^{12}) ^{5}_{(tp)} x Plank time factor _{(sec/tp)} ÷ Julian year _{(sec/yr) } = Age of Universe(yrs)
5 Ø (10)^{60} x 5.39106(32)x10 ^{44} ÷ (60x60x24x365.25) = 13.82065(85) x 10^{9}
Conclusion:
God's Hand precisely measured off our Spacetime heaven!