Question 4040: Sum of Two Integer Principal Distances

Original question and givens

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[42, 56], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

• G001 — In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[42, 56], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-distance-formula

Step-by-step derivation

integer vectors from a common origin;orthogonal equal-length units

L2(v)≥0 and L2(v)²=v₁²+v₂²

vector1 coordinate1 squared := square(42)

vector1 coordinate1 squared = 1764

vector1 coordinate2 squared := square(56)

vector1 coordinate2 squared = 3136

vector1 L2 squared := sum(1764, 3136)

vector1 L2 squared = 4900

vector2 coordinate1 squared := square(5)

vector2 coordinate1 squared = 25

vector2 coordinate2 squared := square(12)

vector2 coordinate2 squared = 144

vector2 L2 squared := sum(25, 144)

vector2 L2 squared = 169

principal root1 square := square(70)

principal root1 square = 4900

root=70≥0;root²=4900=L2²

L2=70

principal root2 square := square(13)

principal root2 square = 169

root=13≥0;root²=169=L2²

L2=13

two distance sum := sum(70, 13)

two distance sum = 83

[70, 13]

sum=83

Reverse checks

• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute two distance sum with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute principal root2 square with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute principal root1 square with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute vector2 L2 squared with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute vector2 coordinate2 squared with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute vector2 coordinate1 squared with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute vector1 L2 squared with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute vector1 coordinate2 squared with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute vector1 coordinate1 squared with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

83

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.