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Prime-Power Extraction, Product Additivity and Sum Cancellation

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations · 2026-10-15 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

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1. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=2. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(2, 2, 2, 1, 1, [2], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 2, 2, 1, 1, [2], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

2. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=-18. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(3, -18, 18, 2, 2, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -18, 18, 2, 2, [3, 9], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

3. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=375. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(5, 375, 375, 3, 3, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 375, 375, 3, 3, [5, 25, 125], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

4. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=-8. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(2, -8, 8, 3, 1, [2, 4, 8], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -8, 8, 3, 1, [2, 4, 8], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

5. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=45. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(3, 45, 45, 2, 5, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 45, 45, 2, 5, [3, 9], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

6. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=-750. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(5, -750, 750, 3, 6, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -750, 750, 3, 6, [5, 25, 125], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

7. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=14. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(2, 14, 14, 1, 7, [2], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 14, 14, 1, 7, [2], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

8. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=-9. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(3, -9, 9, 2, 1, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -9, 9, 2, 1, [3, 9], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

9. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=250. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(5, 250, 250, 3, 2, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 250, 250, 3, 2, [5, 25, 125], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

10. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=-6. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(2, -6, 6, 1, 3, [2], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -6, 6, 1, 3, [2], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

11. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=36. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(3, 36, 36, 2, 4, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 36, 36, 2, 4, [3, 9], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

12. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=-625. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(5, -625, 625, 4, 1, [5, 25, 125, 625], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -625, 625, 4, 1, [5, 25, 125, 625], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

13. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=12. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(2, 12, 12, 2, 3, [2, 4], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 12, 12, 2, 3, [2, 4], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

14. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=-63. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(3, -63, 63, 2, 7, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -63, 63, 2, 7, [3, 9], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

15. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=125. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(5, 125, 125, 3, 1, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 125, 125, 3, 1, [5, 25, 125], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

16. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=-4. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(2, -4, 4, 2, 1, [2, 4], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -4, 4, 2, 1, [2, 4], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

17. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=27. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(3, 27, 27, 3, 1, [3, 9, 27], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 27, 27, 3, 1, [3, 9, 27], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

18. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=-500. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(5, -500, 500, 3, 4, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -500, 500, 3, 4, [5, 25, 125], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

19. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=10. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(2, 10, 10, 1, 5, [2], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 10, 10, 1, 5, [2], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

20. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=-54. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(3, -54, 54, 3, 2, [3, 9, 27], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -54, 54, 3, 2, [3, 9, 27], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

21. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=875. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(5, 875, 875, 3, 7, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 875, 875, 3, 7, [5, 25, 125], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

22. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=-2. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(2, -2, 2, 1, 1, [2], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -2, 2, 1, 1, [2], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

23. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=18. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(3, 18, 18, 2, 2, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 18, 18, 2, 2, [3, 9], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

24. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=-375. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(5, -375, 375, 3, 3, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -375, 375, 3, 3, [5, 25, 125], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

25. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=8. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation

(2, 8, 8, 3, 1, [2, 4, 8], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 8, 8, 3, 1, [2, 4, 8], Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate

26. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=2, integers a=1, b=-4, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(2, 1, -4, 0, 2, -4, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 1, -4, 0, 2, -4, 2, 2, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

27. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=3, integers a=6, b=-27, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(3, 6, -27, 1, 3, -162, 4, 4, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 6, -27, 1, 3, -162, 4, 4, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

28. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=5, integers a=75, b=-4, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(5, 75, -4, 2, 0, -300, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 75, -4, 2, 0, -300, 2, 2, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

29. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=7, integers a=4, b=-35, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(7, 4, -35, 0, 1, -140, 1, 1, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 4, -35, 0, 1, -140, 1, 1, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

30. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=11, integers a=55, b=-726, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(11, 55, -726, 1, 2, -39930, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 55, -726, 1, 2, -39930, 3, 3, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

31. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=13, integers a=1014, b=-7, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(13, 1014, -7, 2, 0, -7098, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 1014, -7, 2, 0, -7098, 2, 2, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

32. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=2, integers a=7, b=-16, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(2, 7, -16, 0, 4, -112, 4, 4, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 7, -16, 0, 4, -112, 4, 4, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

33. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=3, integers a=24, b=-81, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(3, 24, -81, 1, 4, -1944, 5, 5, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 24, -81, 1, 4, -1944, 5, 5, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

34. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=5, integers a=225, b=-10, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(5, 225, -10, 2, 1, -2250, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 225, -10, 2, 1, -2250, 3, 3, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

35. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=7, integers a=10, b=-77, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(7, 10, -77, 0, 1, -770, 1, 1, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 10, -77, 0, 1, -770, 1, 1, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

36. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=11, integers a=121, b=-1452, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(11, 121, -1452, 2, 2, -175692, 4, 4, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 121, -1452, 2, 2, -175692, 4, 4, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

37. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=13, integers a=2028, b=-13, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(13, 2028, -13, 2, 1, -26364, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 2028, -13, 2, 1, -26364, 3, 3, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

38. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=2, integers a=13, b=-28, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(2, 13, -28, 0, 2, -364, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 13, -28, 0, 2, -364, 2, 2, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

39. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=3, integers a=42, b=-135, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(3, 42, -135, 1, 3, -5670, 4, 4, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 42, -135, 1, 3, -5670, 4, 4, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

40. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=5, integers a=375, b=-16, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(5, 375, -16, 3, 0, -6000, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 375, -16, 3, 0, -6000, 3, 3, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

41. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=7, integers a=16, b=-119, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(7, 16, -119, 0, 1, -1904, 1, 1, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 16, -119, 0, 1, -1904, 1, 1, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

42. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=11, integers a=187, b=-2178, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(11, 187, -2178, 1, 2, -407286, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 187, -2178, 1, 2, -407286, 3, 3, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

43. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=13, integers a=3042, b=-19, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(13, 3042, -19, 2, 0, -57798, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 3042, -19, 2, 0, -57798, 2, 2, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

44. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=2, integers a=19, b=-40, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(2, 19, -40, 0, 3, -760, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 19, -40, 0, 3, -760, 3, 3, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

45. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=3, integers a=60, b=-189, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(3, 60, -189, 1, 3, -11340, 4, 4, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 60, -189, 1, 3, -11340, 4, 4, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

46. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=5, integers a=525, b=-22, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(5, 525, -22, 2, 0, -11550, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 525, -22, 2, 0, -11550, 2, 2, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

47. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=7, integers a=22, b=-161, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(7, 22, -161, 0, 1, -3542, 1, 1, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 22, -161, 0, 1, -3542, 1, 1, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

48. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=11, integers a=253, b=-2904, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(11, 253, -2904, 1, 2, -734712, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 253, -2904, 1, 2, -734712, 3, 3, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

49. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=13, integers a=4056, b=-25, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(13, 4056, -25, 2, 0, -101400, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 4056, -25, 2, 0, -101400, 2, 2, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

50. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=2, integers a=25, b=-52, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation

(2, 25, -52, 0, 2, -1300, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 25, -52, 0, 2, -1300, 2, 2, Yes, infinity_plus_k_equals_infinity).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate

51. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=2, nonzero summands a=2, b=4; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(2, 2, 4, 1, 2, 6, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 2, 4, 1, 2, 6, 1, 1, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

52. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=3, nonzero summands a=18, b=27; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(3, 18, 27, 2, 3, 45, 2, 2, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 18, 27, 2, 3, 45, 2, 2, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

53. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=5, nonzero summands a=375, b=4; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(5, 375, 4, 3, 0, 379, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 375, 4, 3, 0, 379, 0, 0, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

54. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=7, nonzero summands a=28, b=35; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(7, 28, 35, 1, 1, 63, 1, 1, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 28, 35, 1, 1, 63, 1, 1, Yes, No, not_guaranteed).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

55. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=11, nonzero summands a=605, b=726; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(11, 605, 726, 2, 2, 1331, 3, 2, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 605, 726, 2, 2, 1331, 3, 2, Yes, No, not_guaranteed).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

56. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=13, nonzero summands a=13182, b=7; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(13, 13182, 7, 3, 0, 13189, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 13182, 7, 3, 0, 13189, 0, 0, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

57. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=2, nonzero summands a=14, b=16; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(2, 14, 16, 1, 4, 30, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 14, 16, 1, 4, 30, 1, 1, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

58. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=3, nonzero summands a=72, b=81; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(3, 72, 81, 2, 4, 153, 2, 2, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 72, 81, 2, 4, 153, 2, 2, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

59. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=5, nonzero summands a=1125, b=10; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(5, 1125, 10, 3, 1, 1135, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 1125, 10, 3, 1, 1135, 1, 1, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

60. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=7, nonzero summands a=70, b=77; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(7, 70, 77, 1, 1, 147, 2, 1, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 70, 77, 1, 1, 147, 2, 1, Yes, No, not_guaranteed).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

61. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=11, nonzero summands a=1331, b=1452; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(11, 1331, 1452, 3, 2, 2783, 2, 2, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 1331, 1452, 3, 2, 2783, 2, 2, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

62. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=13, nonzero summands a=26364, b=13; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(13, 26364, 13, 3, 1, 26377, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 26364, 13, 3, 1, 26377, 1, 1, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

63. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=2, nonzero summands a=26, b=28; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(2, 26, 28, 1, 2, 54, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 26, 28, 1, 2, 54, 1, 1, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

64. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=3, nonzero summands a=126, b=135; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(3, 126, 135, 2, 3, 261, 2, 2, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 126, 135, 2, 3, 261, 2, 2, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

65. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=5, nonzero summands a=1875, b=16; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(5, 1875, 16, 4, 0, 1891, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 1875, 16, 4, 0, 1891, 0, 0, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

66. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=7, nonzero summands a=112, b=119; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(7, 112, 119, 1, 1, 231, 1, 1, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 112, 119, 1, 1, 231, 1, 1, Yes, No, not_guaranteed).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

67. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=11, nonzero summands a=2057, b=2178; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(11, 2057, 2178, 2, 2, 4235, 2, 2, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 2057, 2178, 2, 2, 4235, 2, 2, Yes, No, not_guaranteed).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

68. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=13, nonzero summands a=39546, b=19; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(13, 39546, 19, 3, 0, 39565, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 39546, 19, 3, 0, 39565, 0, 0, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

69. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=2, nonzero summands a=38, b=40; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(2, 38, 40, 1, 3, 78, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 38, 40, 1, 3, 78, 1, 1, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

70. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=3, nonzero summands a=180, b=189; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(3, 180, 189, 2, 3, 369, 2, 2, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 180, 189, 2, 3, 369, 2, 2, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

71. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=5, nonzero summands a=2625, b=22; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(5, 2625, 22, 3, 0, 2647, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 2625, 22, 3, 0, 2647, 0, 0, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

72. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=7, nonzero summands a=154, b=161; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(7, 154, 161, 1, 1, 315, 1, 1, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 154, 161, 1, 1, 315, 1, 1, Yes, No, not_guaranteed).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

73. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=11, nonzero summands a=2783, b=2904; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(11, 2783, 2904, 2, 2, 5687, 2, 2, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 2783, 2904, 2, 2, 5687, 2, 2, Yes, No, not_guaranteed).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

74. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=13, nonzero summands a=52728, b=25; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(13, 52728, 25, 3, 0, 52753, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 52728, 25, 3, 0, 52753, 0, 0, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

75. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=2, nonzero summands a=50, b=52; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation

(2, 50, 52, 1, 2, 102, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 50, 52, 1, 2, 102, 1, 1, Yes, Yes, Yes).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate

76. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=2, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(zero_convention, 2, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 2, 0, infinity, single_serializable_zero_convention).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

77. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=6, integer a=2. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(nonprime_base, 6, 2, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 6, 2, No, not_called_p_adic_valuation).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

78. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=2, integer a=-12. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(negative_normalization, 2, -12, 12, 2, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 2, -12, 12, 2, Yes, negative_integer_is_valid).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

79. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=2, integer a=8. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(equal_valuations_cancellation, 2, 8, -8, 3, 3, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 2, 8, -8, 3, 3, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

80. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=3, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(zero_convention, 3, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 3, 0, infinity, single_serializable_zero_convention).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

81. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=12, integer a=6. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(nonprime_base, 12, 6, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 12, 6, No, not_called_p_adic_valuation).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

82. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=3, integer a=-7. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(negative_normalization, 3, -7, 7, 0, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 3, -7, 7, 0, Yes, negative_integer_is_valid).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

83. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=3, integer a=24. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(equal_valuations_cancellation, 3, 24, -24, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 3, 24, -24, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

84. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=5, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(zero_convention, 5, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 5, 0, infinity, single_serializable_zero_convention).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

85. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=18, integer a=10. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(nonprime_base, 18, 10, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 18, 10, No, not_called_p_adic_valuation).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

86. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=5, integer a=-55. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(negative_normalization, 5, -55, 55, 1, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 5, -55, 55, 1, Yes, negative_integer_is_valid).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

87. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=5, integer a=60. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(equal_valuations_cancellation, 5, 60, -60, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 5, 60, -60, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

88. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=7, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(zero_convention, 7, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 7, 0, infinity, single_serializable_zero_convention).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

89. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=24, integer a=14. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(nonprime_base, 24, 14, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 24, 14, No, not_called_p_adic_valuation).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

90. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=7, integer a=-735. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(negative_normalization, 7, -735, 735, 2, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 7, -735, 735, 2, Yes, negative_integer_is_valid).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

91. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=7, integer a=112. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

92. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=11, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(zero_convention, 11, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 11, 0, infinity, single_serializable_zero_convention).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

93. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=28, integer a=18. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(nonprime_base, 28, 18, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 28, 18, No, not_called_p_adic_valuation).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

94. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=11, integer a=-19. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(negative_normalization, 11, -19, 19, 0, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 11, -19, 19, 0, Yes, negative_integer_is_valid).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

95. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=11, integer a=220. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(equal_valuations_cancellation, 11, 220, -220, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 11, 220, -220, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

96. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=13, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(zero_convention, 13, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 13, 0, infinity, single_serializable_zero_convention).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

97. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=34, integer a=22. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(nonprime_base, 34, 22, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 34, 22, No, not_called_p_adic_valuation).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

98. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=13, integer a=-299. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(negative_normalization, 13, -299, 299, 1, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 13, -299, 299, 1, Yes, negative_integer_is_valid).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

99. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=13, integer a=312. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(equal_valuations_cancellation, 13, 312, -312, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 13, 312, -312, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

100. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=17, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation

(zero_convention, 17, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 17, 0, infinity, single_serializable_zero_convention).

Reviewed step-by-step derivation

p-adic valuation of integers · qf-mod49-pval-boundary · candidate

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